fishfry

Comments

On Gödel's Philosophy of Mathematics
Yes; that's what I said. — Banno

Perhaps I misunderstood. You said:

The argument seems to be that a system's being true (about bridges) is on a par with its being consistent, in that we can only show either by bringing in something form outside - this being a consequence of the second incompleteness theorem.

I'd be interested to hear if others think this an accurate account of Gödel's thinking. — Banno

But in fact that's an accurate account of a position Gödel is refuting, not thinking. Are we more or less in agreement on that?

In any event, on reading the article you linked, it's sufficiently detailed and somewhat confusing to the point that it's not productive to quote-mine it IMO. It's hard to know who's holding what opinion, Carnap or Gödel. Nor do I see how the second incompleteness theorem refutes the syntactic view. to the extent that I follow the quote mining at all. I think the article would benefit from a more clearly description of who is saying what and who agrees or disagrees with who.

What I do know is that Gödel was a Platonist, and believed that (for example) the continuum hypothesis has a definite truth value. Which would not be consistent with a syntactic view of mathematical truth.
On Gödel's Philosophy of Mathematics
The argument seems to be that a system's being true (about bridges) is on a par with its being consistent, in that we can only show either by bringing in something form outside - this being a consequence of the second incompleteness theorem. — Banno

The article describes Gödel's account of the "syntactic view," as it clearly states. From the article you linked: "The argument uses the Second Incompleteness Theorem[1] to refute the view that mathematics is devoid of content." (My emphasis).

In other words Gödel was describing a particular point of view in order to refute it.

Gödel himself was a Platonist. He believed that every mathematical proposition has an objective truth value, whether or not that truth value can be determined from a particular axiom system or not.

See this article

"In his philosophical work Gödel formulated and defended mathematical Platonism, the view that mathematics is a descriptive science, or alternatively the view that the concept of mathematical truth is objective."

I'd be interested to hear if others think this an accurate account of Gödel's thinking. — Banno

It's an accurate account of his description of a philosophical viewpoint that he did not personally hold, but was describing in order to refute it. Just as if I, a committed globist, described my understanding of flat earth theory.
Is it no longer moral to have kids?
Then how is it ok to impose this situation on any child, let alone your own? — hypericin

I'm reminded of Richard Feynman's great essay, Los Alamos from Below (pdf link). It's about his time working as a low-level scientist at the Los Alamos facility to build the first atomic bomb during WWII. At the end he writes:

"I returned to civilization shortly after that and went to Cornell to teach, and my first impression was a very strange one. I can't understand it anymore, but I felt very strongly then. I sat in a restaurant in New York, for example, and I looked out at the buildings and I began to think, about how much the radius of the Hiroshima bomb damage was and so forth ... how far from here was 34th St? ... All those buildings, all smashed -- and so on. And I would go along and I would see people building a bridge, or they'd be making a new road, and I thought, they're crazy, they just don't understand, they don't understand. Why are they making new things? It's so useless.

But, fortunately, it's been useless for about 30 years now, isn't it? So I've been wrong for 30 years about it being useless making bridges and I'm glad that those other people had the sense to go ahead."

He went on to have two children. One became a philosopher and computer scientist; the other, a photographer.
What happened to "I don’t believe the universe is infinite"?
Seconded. I have a distinct recollection of having written something intelligent. Hate to lose 'em - doesn't happen all that often. — tim wood

LOL. Me too.
Taking from the infinite.
The thing with this type of deception — Metaphysician Undercover

I'm going to let you have the last word. I'm out. But for the record, can you please name the specific individuals involved in this deception? We need their names to hold them accountable at the Stalinist show trials to begin soon. Cantor? Zermelo? Mrs. Zermelo, who was pro choice? Abraham Fraenkel? Should John von Neumann be included? He did invent mathematical economics and worked on the hydrogen bomb, but ... he DID do foundational work in set theory as well. Might was well include him on the list. How about the modern set theorists Solovay, Magidor, Shelah? Or the contemporary ones like Woodin and Hamkins? The modern philosophers of set theory like Quine, Putnam, and especially Maddy? Are they all involved in this deception? Please be specific, we need to know how many cells to reserve at Gitmo.

In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?
— fishfry

Sorry fishfry, but you'll need to do a better job explaining than this. Your truth table does not show me how you draw this conclusion. — Metaphysician Undercover

You don't know material implication? You will find "my" truth table on that page. How deep exactly is your ignorance? "My" truth table? Do you really mean to say you never saw this before? I guess I don't understand how that could be. Honestly, in all your time on this forum and presumably studying the philosophy of math, you never saw basic sentential (aka propositional) logic?

Here you go. https://en.wikipedia.org/wiki/Propositional_calculus

You can have the last word. Though after calling modern mathematics a "deception" and admitting that you are unfamiliar with the truth table for material implication, I don't see how you could top what's gone before.

Thanks for the chat. All the best.
Why is so much allure placed on the female form?
A woman in a bikini always elicits a strong response from men and even non-gay women as well. — Maximum7

But enough about women's beach volleyball! Or is that, butt enough?
Taking from the infinite.
I'm a philosopher, — Metaphysician Undercover

A philosophy crank is more like it. You have zero familiarity with the 20th century literature on the philosophy of set theory. You haven't read Maddy, Quine, or Putnam. You have no interest in learning anything about the philosophy of set theory. When I mentioned to you the other day that Skolem was skeptical of set theory as a foundation for math, you expressed no curiosity and just ignored the remark. Why didn't you ask what his grounds were? After all he was one of the major set theorists of the early 20th century.

You don't seem to be able to engage in logical reasoning. You keep saying things that are demonstrably false, and when corrected, you simply repeat the same mistakes in your next post. You don't seem to understand elementary sentential logic, for example material implication and logical equivalence. About this more in a few paragraphs.

my game is to analyze and criticize the rules of other games. — Metaphysician Undercover

What do you think of the knight move in chess? When pressed, you tried to claim it's "physical" because chess sets are made of atoms. Childish sophistry.

This is a matter of interpretation. If you do not like that, then why are you participating in a philosophy forum? — Metaphysician Undercover

I'm here discussing the philosophy of math. I hope you don't think anything we've discussed is actual math. Do you think mathematicians sit around and talk about whether the empty set exists? And as I say, I know a little bit about the actual philosophical literature on the subject, and you haven't the slightest interest in it.

As much as you, as a mathematician are trying to teach me some rules of mathematics, I as a philosopher am trying to teach you some rules of interpretation. — Metaphysician Undercover

Is that why you can't work your way through a simple logical equivalence in sentential logic?

So the argument goes both ways, you are not progressing very well in developing your capacity for interpreting. — Metaphysician Undercover

Your cranky and ignorant ideas of math aren't subject to interpretation, only derision.

But if you do not like the game of interpretation, then just do something else — Metaphysician Undercover

Oh I'll be here to discuss the philosophy of math, if anyone is interested. Sadly you have nothing intelligible to say on the subject.

This is why the axiom of extensionality is not a good axiom. It states something about the thing referred to by "set", which is inconsistent with the mathematician's use of "set", as you've demonstrated to me. — Metaphysician Undercover

Nonsense. It's your own inability to follow an elementary exercise in logic that keeps you stuck.

We've already been through this problem, a multitude of times. That two things are equal does not mean that they are the same. — Metaphysician Undercover

That's not relevant here. However, two things that are mathematically equal are indeed the same.

That's why I concluded before, that it's not the axiom of extensionality which is so bad, but your interpretation of it is not very good. — Metaphysician Undercover

Mine is the perfectly standard interpretation, comprehensible to everyone who spends a little effort to understand it. Two sets are the same if and only if they have the same elements. Formally, if a thing is in one set if and only if it's in the other; which (as we will shortly see) includes the case where both sets are empty.

But I now see that the axiom of extensionality is itself bad. — Metaphysician Undercover

Like I say, you need to take that up with Ernst Zermelo. Or the authors of every set theory text in the world.

In case you haven't noticed, what I am interested in is the interpretation of symbols. — Metaphysician Undercover

How can that be? You are completely unable to understand even the most elementary symbolic reasoning.

And obviously the symbology of the axiom is not perfectly clear. If you can interpret "=" as either equal to, or the same as, then there is ambiguity. — Metaphysician Undercover

Are you saying that two things can be "the same" but not equal? Are you sure whatever you're on is legal in your jurisdiction?

Actually, I'm starting to see that this, what you claim in your vacuous argument, is not a product of the axiom of extensionality, but a product of your faulty interpretation. By the axiom of extensionality, a person on the moon is equal to a pink flying elephant, and you interpret this as "the same as". So the axiom is bad, in the first place, for the reasons I explained in the last post, and you make it even worse, with a bad interpretation. — Metaphysician Undercover

I'm going to walk you through this.

First, do you understand material implication? Material implication has the following truth table:
```
P    Q    P ==> Q
--------------------
T    T          T
T    F          F
F    T          T
F    F          T
```
In other words "If 2 + 2 = 5 then I am the Pope" is a true material implication. Do you understand that? Do you agree? Do you have a disagreement perhaps?

Ok. Let $A$ be the set of pink flying elephants, and $B$ the set of people on the moon. For purposes of this exercise, let's assume these are not contingent. If you can't do that then make them the set of even numbers not divisible by 2, and the set of primes with nontrivial factorizations if you like.

Now I claim that for all $X$ , it is the case that $X \in A \iff X \in B$ . That is read as, "X is an element of A if and only if X is an element of B.

In sentential logic we break this down into two propositions: (1) If X is an element of A then X is an element of B; and (2) If X is an element of B then X is an element of A."

Now for (1). If X is a pink flying elephant, then it's a person on the moon. Is that true? Well yes. There are no pink flying elephants and there are no people on the moon. So this is line 4 of the truth table, the F/F case, which evaluates to True. So (1) is true.

How about (2)? Well the argument is exactly the same. If X is a person on the moon, then they are a pink flying elephant. As in (1), this is the False/False case of material implication.

Having shown both directions of the implication, we have that A and B are logically equivalent.

The axiom of extensionality says that if it's the case that X is in A if and only if X is in B, then A = B.

Therefore, by an exercise in elementary sentential logic, we see that the axiom of extensionality says that the two sets are the same.

Now here is what I know. I know that you are totally incapable of following this simple chain of logic. Or perhaps just unwilling. Either way, it's no longer my problem.

You really do not seem to be getting it. If, we can "use a predicate to form a set" as the axiom of specification allows, then it is not true that a set is characterized by its elements. — Metaphysician Undercover

Of course it is. We can use specification to prove the existence of some set; AND by extension if that set has the same elements as some other set, then the two sets are the same.

It's characterized by that predication. — Metaphysician Undercover

No no no. "Characterized" in this context is YOUR word but it's not what I've said and not what set theory says. Set theory says that (1) we can show some set exists using specification; and that set is also subject to extensionality: if some other set has the same elements, then the two sets are the same.

For example the set of "the first three positive integers is {1,2,3}. The set of the positive square roots of 1, 4, and 9 is (1, 2, 3}. Two distinct specifications. But the two sets have exactly the same elements; so they are the same set.

Specification lets you show some set exists. Extensionality tells you when some set is equal to some other set.

I can not for the life of me figure out why you won't get this. I will say that I'm not sure how we got onto this particular subtopic, and that I'm finding it tedious in the extreme, so I probably won't be replying back much as long as you continue to be trollishly repeating this fallacious line of argument.

The two are mutually exclusive, inconsistent and incompatible. — Metaphysician Undercover

No, they are quite independent of each other. Specification is one way (out of several others) to show that a given set exists. Extensionality tells you when that set is equal to some other set.

Specification allows for a nonempty set, I have no problem with this. But to say that this set is characterized by its elements is blatantly false. It has no elements, and it is characterized as having zero elements, an empty set. So it's not characterized by its elements, it's characterized by the number of elements which it has, none. . — Metaphysician Undercover

I just gave you a formal proof to the contrary. An object is in A if and only if it's in B. If A and B happen to be empty, that is a true statement. Therefore by extensionality, A = B. Or in words, "There's only one empty set."

Yes, this is the problem with the axiom of extension, in its portrayal of the empty set. It is saying that if two specified sets each have zero elements, then "the elements themselves" are equal. — Metaphysician Undercover

You're confusing yourself because you are unable/unwilling to do a little basic sentential logic, or perhaps that you do not understand material implication.

Which is it?

However, there are no such elements to allow one to judge the equality of them. — Metaphysician Undercover

You are confusing yourself because you can't/won't follow the symbolic logic. For all X, X is in A if and only if x is in B. That's as clear as can be.

Tell me, do you disagree or happen to not know basic sentential logic? Do you understand that "if 2 + 2 = 5 then I am the Pope" is true?

So there is no judgement — Metaphysician Undercover

No middle e in judgment. Please make a note of it.

that "the elements themselves" are equal, because there are no elements to judge, and so the judgement of cardinal equivalence, that they have the same number of elements, zero, is presented as a judgement of the elements themselves. — Metaphysician Undercover

X is in A if and only if X is in B. Can it be the case that you don't actually know basic sentential logic, and don't understand material implication, and therefore cannot understand the content of the axiom of extensionality? That would explain a lot.

You ought to recognize, that to present a judgement of cardinal equivalence, as a judgement of the elements themselves, is an act of misrepresentation, which is an act of deception. I know that you have no concern for truth or falsity in mathematical axioms, but you really ought to have concern for the presence of deception in axioms. — Metaphysician Undercover

You interpret your own ignorance as deception by others. Pretty funny.

Now, do you agree, that when there are no elements, it makes no sense to say that the elements themselves are respectively equal? — Metaphysician Undercover

They are vacuously respectively equal. I see that it must be the case that you do not actually understand material implication. That explains quite a lot.

The axiom of extensionality, X is in A if and only if X is in B, is the same as "If 2 + 2 = 5 then I am the Pope; AND if I am the Pope then 2 + 2 = 5." I see that you truly don't get this. @Meta my friend you need to go study up on basic sentential logic. As a self-proclaimed philosopher you are missing the very basics.

What is really being judged as equal is the cardinality. They both have zero elements. — Metaphysician Undercover

I see why you think this. It's because you don't understand material implication and logical equivalence.

No, the axiom of extensionality does not tell us when two sets are the same, that's the faulty interpretation I've pointed out to you numerous times already, and you just cannot learn. It tells us when two sets are equal. — Metaphysician Undercover

Someone a while back pointed out that in math, equality is extensional. You are taking it intentionally. But whatever. It's not relevant to the fact that we've discovered that you can't do basic sentential logic. That's something you should remedy at once.

That faulty interpretation is what enables the deception. Equality always indicates a judgement of predication, and in mathematics it's a judgement of equal quantity, which you call cardinal equivalence. When you replace the determination of the cardinality of two empty sets, "equal", with "the same", you transfer a predication of the set, its cardinality, to make a predication of its elements, "the same as each other". I believe that's known as a fallacy of division. — Metaphysician Undercover

Throwing more crap on the wall doesn't remedy your inability to understand a simple logical argument.

Well, "pink flying elephants" was your example, and it's equally contingent. — Metaphysician Undercover

For purposes of this discussion, we take the two predicates as absolute and not contingent. You're just raising this red herring to sow confusion. The only one confused here is you.

The issue of temporally contingent propositions raises a completely different problem. The only truly necessary empty set is the one specified as "the empty set". As your examples of square circles and married bachelors show, definitions and conceptual structures change over time, so your assertion that mathematics has no temporally contingent propositions is completely untrue. It may be the case that "the empty set" will always refer to the empty set, necessarily, but how we interpret "empty" and "set" is temporally contingent. So temporal contingency cannot be removed from mathematics as you claim. This is the problem of Platonic realism, the idea that mathematics consists of eternal, unchanging truths, when in reality the relations between symbols and meaning evolves. — Metaphysician Undercover

You can throw all the crap on the wall you like. I'm done cleaning it up. Change the subject, I'm done with this. Go read a book on logic and then work through the axiom of extensionality, which frankly is quite simple.
Driving the automobile is a violation of civic duty.
may indeed be for the elite — Sha'aniah

As opposed to the rest of us ignorant cattle. It's striking how often these kinds of lofty sentiments eventually come down to raw elitism. That's why we have the white liberal elites living in gated communities and calling for the abolition of police, while actual black residents of crime-ridden neighborhoods overwhelmingly want more police.

Let's get rid of the Wal Mart and install 5 small grocery outlets — Sha'aniah

In 2020 the US government wiped out local small businesses and drove record profits of Amazon and WalMart. An unprecedented transfer of wealth from the middle classes to the rich. Aided and abetted as usual by the liberal elite, who know so much better than the rest of us how the rest of us should live.

If you want to know why there's a resurgence of worldwide populism, this is why. The elite talk a good game while promoting massive inequality to further enrich themselves. The Obamas preach global warming and buy expensive beachfront property. Just look at the carbon footprint of the jet-sitting climatistas. Jennifer Lopez and Ben Affleck vacation aboard a 140 foot mega-yacht that uses more energy than some third-world nations. And so on.

Not attacking your idealism, which is commendable if naive. But far too often, if you look at the holier-than-thou rhetoric among the most greedy and rapacious members of society, you see that the more pious the social commentary, the uglier the soul.
Driving the automobile is a violation of civic duty.
Please do read the essay. I tried to make it free but amazon is a business.

I understand 100 yrs ago needing a car. But not anymore. So much more is known about societal and cultural options there's no reason to drive away anymore. Each small town can be its own little city. I dont reject highway travel. Its simply neglectful to isolate yourself in a small community. Its poor application of spontaneity and person to person commerce due to how much weight and how many obligations the automobile carries. A person needs to walk in off the street, not drive there. If you drive there, you dont know them. Not the way you know them if you sleep there.

The small town now has so much potential. Yoga, martial arts, different philosophical perspectives, cuisine from all over the world. As long as we have time and space through which to do these things.

Many people try to live off the state by paying more for cars and insurance instead of eating essential and having social and dietary needs met. Then they work 60 hrs a week to afford daycare for kids they dont raise. GREAT. — Sha'aniah

Do you like James Howard Kunstler? He's a proponent of downsized living. His novel is called, "A World Made By Hand." Good essayist. His main thesis is "the long emergency," meaning that the world's running out of oil and we're all going to have to make some new arrangements, of the downsizing variety.

I agree with your ideals, but the practicality is tricky. I live in a relatively small town but the grocery stores and other services are not within walking distance.
Arguments for livable minimum wage.
now I get the joy of still paying my bills AND the deadbeats' bills because they stopped paying theirs. Just awesome. — Book273

Yup. That's how it works. The wealthy can buy politicians and lawyers. The poor have no money. That leaves the working stiffs to pay for it all.
Taking from the infinite.
But political neutrality doesn't amount to reasonableness regarding which mathematics should be prioritised. — sime

Which math gets prioritized is of course a matter of historical contingency. Set theory in the 20th century, maybe category theory / type theory / topos theory / whatever in the 21st, and who knows what in the 22nd. I agree with you about this. However, you have expressed an antipathy to the axiom of choice that is not shared by category theorists. I'm not exactly sure where you are coming from. Are you a constructivist? They are gaining mindshare these days through the influence of computers. But Turing showed us that there are easily-stated problems that can not possibly be solved by a computer. So there will always be a place for nonconstructive math. In my "inappropriately trained" opinion, of course.

to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis. — sime

I got a chuckle out of this.
Taking from the infinite.
You do not seem to be grasping the problem. — Metaphysician Undercover

Rather than try to understand set theory on its own terms, you just want to fight with it. Why? I'm taking the trouble to explain it to you, on its own terms. I'm on record multiple times saying that I make no claims that it's "true" in any meaningful sense; or even meaningful in any meaningful sense. So why try to argue with me about the subject? The person you want to address your complaints to is [url=https://en.wikipedia.org/wiki/Ernst_ZermeloErnst Zermelo[/url], who more than anyone is responsible for the modern incarnation of the standard axioms of set theory. Cantor gets all the credit and Zermelo did the heavy lifting. Zermelo died in 1953 so he's not available for you to complain to; but I am not available for you to complain to either. I"m not defending the truth, meaning, sanity, or sense of set theory. I'm only describing to you how it is. You will have to take your complaints elsewhere.

Perhaps this is a fundamental confusion on your part, but I don't see why. I have explained my viewpoint many times. I'm describing set theory to you. I am not defending it, not advocating it, not promoting it. But I am, to the best of my ability, giving an accurate account of the basics of the subject as it is understood by mathematicians. So if you want to learn it, let's proceed. If you just want to argue with me about it, that makes for a one-sided and tedious interaction.

Let me say this once and for all: I am not the lord-high defender of set theory. It's exactly like chess. I'm teaching you the rules. If you don't like the game, my response is for you to take up some other game more to your liking.

If a set is characterized by its elements, there is no such thing as an empty set. — Metaphysician Undercover

You're too hung up on the empty set. As I said in my previous post, students generally have a hard time getting accustomed to vacuous arguments. It's perfectly analogous to the subject of material implication in logic. Students don't get why "If 2 + 2 = 5 then I am the Pope" is a true statement. At some point most of them get it, and some never do.

No elements, no set. Do you understand this? — Metaphysician Undercover

No. It's not only that you're wrong, but it's a nothingburger of an issue. It's like a beginning logic student constantly arguing with the professor about material implication. The facts of the matter are never going to change. The student can only accept it, or drop the course and enroll in a different one more to his or her liking. One tactic is to just accept it ("it" being whatever is bothering the learner at the moment) on faith, keep working at the subject, and one they they'll wake up and realize that it's all perfectly obvious and they can't even remember a time when it wasn't. That's the tactic I recommend to you.

That is the logical conclusion we can draw from " a set is characterized by its elements". — Metaphysician Undercover

But I have already explained to you in my previous post, that "a set is characterized by its elements" is merely an English-language approximation to the axiom of extentionality, which actually says, $\forall A \forall B (\forall X (X \in A \iff X \in B) \implies A = B)$ . That is the axiom that says that two sets are equal if they have exactly the same elements. And by a vacuous argument -- the same kind of argument that students have had trouble with since logic began -- two sets are the same if they each have no elements.

If you won't grapple with the symbology, you have to accept it on faith. You can't just fall back on the imprecise English-language version, now that I've shown you (twice) the formal version.

If we have no elements, we have no set. If you do not agree with this, explain to me how there could be a set which is characterized by its elements, and it has no elements. It has no character? Isn't that the same as saying it isn't a set? — Metaphysician Undercover

You're being tedious. First, the matter is trivial. The empty set is a thing in set theory. You can't allow your learning to be stuck on this one point. Accept it and move on, or go find something else to be interested in. Secondly, if you will put in the work to understand the symbology as described on the Wiki page for the axiom of extensionality, at some point you'll probably just get it.

So we cannot proceed to even talk about an empty set because that's incoherent, unless we dismiss this idea that a set is characterized by its elements. — Metaphysician Undercover

The formal symbolic expression of extensionality could not be more clear. The fact that you won't engage with it is not my problem.

Can we get rid of that idea? — Metaphysician Undercover

Perhaps. But again, you're trying to learn a subject, and every time you're shown one of the basic principles, you just want to argue. You make it difficult on yourself. Once you learn basic set theory, you can set about developing an alternative version if you like. Einstein changed physics, but before he did that he mastered classical physics. Right? Right.

Then we could proceed to investigate your interpretation of the axiom of extensionality, which allows you to say "If you have two sets such that they have no elements, they're the same set; namely the empty set", because "empty set" would be a coherent concept. Until we get rid of that premise though, that a set is characterized by its elements there is no such thing as a set with no elements, because such a set would have no identity whatsoever, and we could not even call it a set. — Metaphysician Undercover

You're just being tedious. The formal symbology is perfectly clear. And even if it isn't clear to you, you should just accept the point and move on, so that we can discuss more interesting things. You're causing yourself to get stuck on a relatively minor point. You have two ways out: One, grapple with the formal symbology here. Two, accept it and move on. Repeating the same tired and fallacious objections is no longer an option, at least with me. Maybe you can get someone else to play.

You are not grasping the distinction between 'characterized by its elements', and 'characterized by its specification' which I'm trying to get though to you. — Metaphysician Undercover

Here you have made up your own phrase, "characterized by its specification." I have not said that. The axiom of specification is one of the axioms that tell us when particular sets may be said to exist. I explained this to you in painful detail in my previous post, and you are just ignoring what I said. The axiom of extensionality tells us when two sets are the same. The axiom schema of specification tells us when certain sets exist. There are other set existence axioms: pairing, union, intersection, powerset, replacement, and choice. Frankly if you want to complain about axioms, it's replacement you should be concerned about. It's very murky. Zermelo's original formulation didn't even include it, as I've recently learned due to @TonesInDeepFreeze's repeated mention of Zermelo set theory, or Z. I went and looked it up and learned something new.

When you say "the set of all purple flying elephants", this is a specification, and this set is characterized by that specification. There are no elements being named, or described, and referred to as comprising that set, there is only a specification which characterizes the set. — Metaphysician Undercover

If by specification you mean predicate, then "E(x) = x is a flying elephant" is most definitely a specification. And in the axiom of specification, that's what's meant. The fact that a predicate may have an empty extension is not a bug, it's a feature.

Every set is entirely characterized by its elements.
— fishfry

Where do you get this idea from? — Metaphysician Undercover

From the axiom of extensionality. You know, I don't care if you believe in the empty set or not. But after my having directedyour attention to the axiom of extensionality so many times, I don't see how you can ask where I got the idea. It's one of the axioms.

Clearly your example "the set of all purple flying elephants" is not characterized by its elements. You have made no effort to take elements, and compose a set You have not even found any of those purple flying elephants. In composing your set, you have simply specified "purple flying elephants". Your example set is characterized by a specification, not by any elements. If you do not want to call this "specification", saving that term for some special use, that's fine, but it's clearly false to say that such a set is characterized by its elements. — Metaphysician Undercover

Accept it and let's move on; or don't accept it and quietly seethe while we move on. I can't engage with you on this point anymore. I've already explained it. The axiom of specification allows us to use a predicate to form a set. The predicate is not required to have a nonempty extension.

This is what happens when we proceed deep into the workings of the imagination. We can take a symbol, a name like "purple flying elephants", or any absurdity, or logical incoherency, like "square circles", each of which we assume has no corresponding objects — Metaphysician Undercover

As I'm always fond of pointing out, the unit circle in the taxicab metric is a square. There's a picture of a square circle at this link. How about married bachelor, that's a better example.

However, we can then claim something imaginary, a corresponding imaginary object, and we can proceed under the assumption that the name actually names something, a purple flying elephant in the imagination. You might then claim that this imaginary thing is an element which characterizes the set. But if you then say that the set is empty, you deny the reality of this imaginary thing, and you are right back at square one, a symbol with nothing corresponding. And so we cannot even call this a symbol any more, because it represents nothing. — Metaphysician Undercover

It's merely a predicate with an empty extension.

Now you've hit the problem directly head on. To be able to have an empty set, a set must be characterized by it's specification, as I've described, e.g. "pink flying elephants". So. the set of pink flying elephants is one set, characterized by the specification "pink flying elephants", and the set of people on the moon is another set, characterized by the specification "people on the moon". To say that they are exactly the same set, because they have the same number of elements, zero, is nor only inconsistent, but it's also a ridiculous axiom. — Metaphysician Undercover

What's true is that given any thing whatsoever, that thing is a pink flying elephant if and only if it's a person on the moon. So the axiom of extensionality is satisfied and the two sets are equal. If you challenged yourself to work through the symbology of the axiom of extentionality this would be perfectly clear to you.

Would you say that two distinct sets, with two elements, are the exact same set just because they have the same number of elements? — Metaphysician Undercover

Of course not. It's not a matter of cardinal equivalence. The elements themselves have to be respectively equal. {1,2} and {1,2} are the same set. {1,2} and {3,47} are not.

I think you'll agree with me that this is nonsense. — Metaphysician Undercover

It would be false. It would not be nonsense, that's a value judgment. And your value judgments regarding mathematics are not good.

And to say that each of them has the very same elements because they don't have any, is clearly a falsity because "pink flying elephants" is a completely different type of element from "people on the moon". If at some point there is people on the moon, then the set is no longer empty. — Metaphysician Undercover

You are right about that. But that's because we are making up examples from real life. Math doesn't have time or contingency in it. 5 is an element of the set of prime numbers today, tomorrow, and forever. The "people on the moon" example was yours, not mine. I could have and in retrospect should have objected to it at the time, because of course it is a temporally contingent proposition. I let it pass. So let me note for the record that there are no temporally contingent propositions in math.

But the two sets have not changed, they are still the set of pink flying elephants, and the set of people on the moon, as specified, only membership has changed. Since the sets themselves have not changed only the elements have, then clearly they were never the same set in the first place. — Metaphysician Undercover

It's a bad example because one of your propositions is temporally contingent. I noted that at the time you mentioned it but let it pass. I see that was a mistake. I have now rectified my error. There are no temporally contingent propositions in math.

Of course, you'll claim that a set is characterized by its elements, so it was never "the set of pink flying elephants in the first place, it was the empty set. But this is clearly an inconsistency because "pink flying elephants was specified first, then determined as empty. So that is not how you characterized these sets. You characterized them as "the set of pink flying elephants", and "the set of people on the moon". — Metaphysician Undercover

The two sets, assuming that we mean at the present moment, are the same, namely the empty set, because the condition in the axiom of extensionality is satisfied. A thing is in one of those sets if and only if it's in the other. Therefore the sets are the same. That's all there is to it.

If you had specified "the empty set", then obviously the empty set is the same set as the empty set, but "pink flying elephants", and "people on the moon" are clearly not both the same set, just because they both happen to have zero elements. The emptiness of these two sets is contingent, whereas the emptiness of "the empty set" is necessary, so there is a clear logical difference between them. — Metaphysician Undercover

Only by virtue of the people on the moon being temporally contingent. So it's a bad example, which I should have pointed out when you first mentioned it.

I don't know why you can't see this as a ridiculous axiom. You say that a "person on the moon" is a "pink flying elephant". That's ridiculous. — Metaphysician Undercover

Nobody says that. What is true is that the axiom of extensionality is satisfied. Until you roll up your sleeves and put in the work to understand that, you'll spin yourself in circles.

See the consequences of that ridiculous axiom? — Metaphysician Undercover

You are being tedious. You wrote an entire post on the nonsense. Go understand what the axiom of extensionality says. You're running yourself in circles because you can't be bothered to challenge yourself to work through what the axiom says.

Now you are saying that a pink flying elephant is a thing which is not equal to a pink flying elephant, and a person on the moon is not equal to a person on the moon. Face the facts, the axiom is nonsensical. — Metaphysician Undercover

Nobody is saying a pink flying elephant is a thing. You're just a logic student having trouble with vacuous arguments. Put in the work to understand it, or just accept it and move on. Being endlessly tedious, writing an entire post about your own misconceptions, is pointless.

Obviously, the axiom of extension is very bad because it fails to distinguish between necessity and contingency. — Metaphysician Undercover

There are no temporally contingent propositions in math. The people on the moon example is a bad one for that reason. I was wrong rhetorically to let it pass without objection earlier, because now you just want to use it to make a sophistic point.

Bottom line is that the empty set is a purely formal object that satisfies some formal conditions. It's not "real" and it's not helpful to try to understand it in terms of common sense. The following SE thread, in particular the checked answer, may be helpful.

https://philosophy.stackexchange.com/questions/14823/why-do-we-have-a-problem-about-understanding-the-concept-of-the-empty-set
Taking from the infinite.
All of which is tantamount to saying that ZF has only partial relevance to modern mathematics in terms of being an axiomatization of well-foundedness, whilst ZFC is completely and utterly useless, failing to axiomatize the most rudimentary notions of finite sets as used in the modern world. — sime

This has murky and unclear relevance to what went before in the same post. I can't tell if you are trying to explain something to me or promoting an agenda. More clarity and less stridence would be helpful to me; but I'm not sure if being helpful is your intention.

What I mean is that one can look up the entry for the axiom of choice on nLab, without encountering a rant against ZFC. So I think you're the one adding that part, and not your fellow constructivists / category theorists / programmers or whatever direction you're coming from.

Indeed, nLab expresses choice as "every surjection splits," which they note means "every surjection has a right inverse," in set theory. This formulation is easily shown to be equivalent to the traditional statement of the axiom of choice. There is no distance between the category-theoretic and set-theoretic views of choice.
Driving the automobile is a violation of civic duty.
Indeed that is the greatest evil of the automobile, that it replaced the horse — Leghorn

As I understand it, cities in the late 1800's had streets covered in horse manure and didn't smell very good in the summer. It's easy to romanticize the past but it must have been awful.
Driving the automobile is a violation of civic duty.
Oh I wish I could be such a jester, but if you want to know what I REALLY think about cars, the UNCENSORED essay is available for viewing on Amazon.

TO WHOM IT MAY CONCERN BY ADAM BRUNSWICK. — Sha'aniah

For sake of conversation, would you have made the same argument in the days of the horse and buggy? Do you object to the "Surrey with a fringe on top?"

What I mean is, do you object to transportation in general? Or just internal combustion engines? If the latter, how about steam engines? Should we bring back the Stanley Steamer?
Taking from the infinite.
It's not trivial, because it's a demonstration of what "specified" means. If you specify that the guests are all human, then clearly that is a specification. If you do not appreciate that specification because it does not provide you with the information you desire, then the specification is faulty in your eyes. But it's false to say that just because you think the specification is faulty, then there is no specification. There is a specification, but it is just not adequate for you. That is simply the nature of specification, it comes in all different degrees of adequacy, depending on what is required for the purpose. But an inadequate specification, for a particular purpose, is in no way a total lack of specification. — Metaphysician Undercover

I've already agreed numerous times that if you insist on your own definition, you're right. You can't convince me that your way of looking at it isn't trivial. I originally understood you to be claiming that the members of every set had something in common that distinguished them of all the non-members. That's an interesting statement, and you'd find many constructivists who agree with you (to the extent that I understand the mindset of the constructivists).

I was surprised -- shocked, in fact -- to discover that you didn't mean that at all, but only meant that you could find some superset that contained the potential candidates for our set. That's a much weaker criterion, I hope you at least agree to that.

So yes, all the guests are human, but that hardly helps the security guard to know who to let in and who to keep out. And every element of the set of primes is a natural number, but that doesn't tell me what's a prime and what's not. It's an incredibly weak criterion. I'm happy to let you have it, but it's trivial because it's such a weak criterion as to be utterly useless in determining which elements are in a given set.

Do you see then, that if "A set is entirely characterized by its elements", then a so-called empty set is not possible? If there are no elements, under that condition, then there is no set. A set is characterized by its elements. There are no elements. Therefore there is no set. If we adhere to this premise, "the set is entirely characterized by its elements", then when there is no elements there is no set. — Metaphysician Undercover

First, the empty set is the unique set that has no elements at all. It's characterized by not having any elements.

But for a more precise answer, we have to look at the actual, exact formal statement of the axiom of extensionality. The natural language version, "A set is entirely characterized by its elements," is just an approximation to what the axiom actually says. The problem is that you don't relate to symbolic reasoning at all. That said, and for the record, I'll walk you through what the axiom of extensionality actually says; and for reference, you can see the Wiki link.

Axiom of extenionality: $\forall A \forall B (X \in A \iff X \in B) \implies A = B$

We unpack this as follows. It says that

For all sets $A$ and $B$ :

If it happens to be the case that for all sets $X$ , $X \in A$ if and only if $X \in B$ ;

Then $A = B$ .

This says in effect that if two sets have exactly the same elements, they're the same set. But the way it's written, it also includes the case of a set with no elements at all. If you have two sets such that they have no elements, they're the same set; namely the empty set.

Now I know this isn't your cup of tea. And that's ok. All you need to know about this is that when you drill down into the technical details of what the axiom of extensionality actually says, the case of the empty set is included.

A more general point is that you want to criticize set theory based on vague natural language descriptions rather than grappling with the actual formal symbology. But in the end, we all have to roll up our sleeves and grapple with the symbology.

Another point is that everyone has trouble with vacuous arguments and empty set arguments. If 2 + 2 = 5 then I am the Pope. Students have a hard time seeing that that's true. The empty set is the set of all purple flying elephants. A set is entirely characterized by its elements; and likewise the empty set is characterized by having no elements. John von Neumann reportedly said, "You don't understand math. You just get used to it." The empty set is just one of those things. You can't use your common sense to wrestle with it, that way lies frustration.

This is logically inconsistent with "a set is entirely characterized by its elements", as I explained in the last post. Either a set is characterized by its elements, or it is characterized by its specified predicates, but to allow both creates the incoherency which I referred to. One allows for an empty set, the other does not. — Metaphysician Undercover

You keep equivocating specification by predicates, on the one hand; and the axiom of extensionality, on the other. Every set is entirely characterized by its elements. Secondly, quite separately from that fact, are various ways of showing the existence of sets. Given a collection of sets we can take their union; or their intersection. Given a set we can take its powerset. Given a set and a predicate we can use the axiom of specification to obtain a subset of the original set consisting of exactly those of its members satisfying the predicate. There's also the axiom of replacement, and the axiom of choice. So first, a set is entirely characterized by its elements. And secondly, we have a toolbox for showing the existence of various sets: union, intersection, powerset, specification, replacement, choice. It's sort of like knowing what a house is, then learning how to build one. There's no ambiguity. Those are two separate things.

We've been through this already. You clearly have referred to the members of the Vitali set. You've said that they are all real numbers. Why do you believe that this is not a reference to the members of the set? You can say "all the people in China", and you are clearly referring to the people in China, but to refer to a group does not require that you specify each one individually. — Metaphysician Undercover

Of course all the members of the Vitali set are real numbers. As are all the elements that are NOT members of the Vitali set. So you can have your point, but it's rather pointless. It doesn't do you any good.

This seems to be where you and I are having our little problem of misunderstanding between us. It involves the difference between referring to a group, and referring to individual. I believe that when you specify a group, "all the guests at the hotel" for example, you make this specification without the need of reference to any particular individuals. You simply reference the group, and there is no necessity to reference any particular individuals. In fact, there might not be any individuals in the group (empty set). You seem to think that to specify a group, requires identifying each individual in that group. — Metaphysician Undercover

Well that's fine, then you believe in nonconstructive sets. You are willing to take the Vitali set and its members at face value. That's great. But you can see that it's very different than the set of prime numbers. With the set of prime numbers, we can look at a given individual number and say, "Yes you're in the club," or "No you're not in the club." With the Vitali set, there is no way to do that.

This is the two distinct, and logically inconsistent ways of using "set" which I'm telling you about. — Metaphysician Undercover

They're not logically inconsistent, they're different ways of building sets; just as using brick or using wood are two different ways of building houses.

We can use "set" to refer to a group of individuals, each one identified, and named as a member of that set (John, Jim, and Jack are the members of this set), or we can use "set" to refer simply to an identified group, "all the people in China". — Metaphysician Undercover

Yes. We can do one or the other. We can talk about the set of prime numbers, in which we can talk about the entire set AND determine exactly which natural numbers are allowed into the set; and we can talk about the Vitali set, where we can NOT determine for any particular real number whether it belongs in the set or not.

Two different ways of obtaining or showing the existence of sets. There are lots of different ways of building houses and lots of different ways of building sets. I don't know why you think this is a problem.

Do you see the logical inconsistency between these two uses, which I am pointing out to you? — Metaphysician Undercover

No, I only see various ways of building sets. Unions, intersections, powersets, specification, replacement, and choice. I believe those are all the set-building or set-existence tools. Like styles of houses or perhaps construction techniques or different choices of materials.

In the first case, if there are no identified, and named individuals, there is no set. Therefore in this usage there cannot be an empty set. — Metaphysician Undercover

The axiom of extensionality provides for the empty set. It's the set with no elements.

But in the second case, we could name the group something like "all the people on the moon", and this might be an empty set. — Metaphysician Undercover

Aha! You're on the verge of getting it. The set of pink flying elephants is an empty set. The set of people on the moon is an empty set. And the axiom of extensionality says that these must be exactly the same set. Because an object is an element of one if and only if it's an element of the other. I hope you'll take a moment to work through the logic. There is only one empty set, because the axiom of extensionality says that if for every object, it's a person on the moon if and only if it's a pink flying elephant, that the two sets must be the same.

I must say, I really do not understand your notation of the empty set. Could you explain? — Metaphysician Undercover

We know from the law of identity that everything is equal to itself. So what is the set of all things that are not equal to themselves? It's the empty set. And by the axiom of extensionality, it's exactly the same as the set of pink flying elephants and the people on the moon.

We notate this as $\{x : x \neq x\}$ . It's read: "The set of all x such that x is not equal to x." As an expert on the law of identity you will agree that there are no such x that satisfy that condition. So we have specified the empty set.

This doesn't help me. — Metaphysician Undercover

I hope my more detailed explanation was better.

Actually you don't seem to be getting my point. The point is that if a set is characterized by its predicates, then an empty set is possible, so I have no problem with "the empty set is the extension of a particular predicate". — Metaphysician Undercover

Ok.

Where I have a problem is if you now turn around and say that a set is characterized by its elements, — Metaphysician Undercover

You're confusing two different things. The axiom of extensionality tells you when two sets are identical: Namely, when they have exactly the same elements.

The axiom of specification, the powerset axiom, the axiom of choice, the axiom schema of replacement, and the axioms of union, intersection, and pairing (I forgot to mention that one) are all axioms that tell us which particular sets exist.

So we have an axiom that tells us when two sets are equal. That's extensionality, or "a set is entirely characterized by its elements." And we have a toolbox of ways to show that various sets exist. The're not in conflict with each other, any more than saying what a house is, is in conflict with the various construction techniques and styles of houses.

because this would be an inconsistency in your use of "set", as explained above. — Metaphysician Undercover

No. The axiom of extensionality tells us when two sets are the same; namely, when they have exactly the same elements. The other axioms are a toolbox for knowing which sets exist. Specification is one of the tools as are union, intersection, etc.

A set characterized by its elements cannot be an empty set, because if there is no elements there is no set. — Metaphysician Undercover

The empty set satisfies the symbolic expression I discussed earlier, whether or not it satisfies the English-language version. I can only refer you to the Wiki page on the axiom of extension, which I keep pointing you to and you keep not reading.

Do you apprehend the difference between "empty set" and "no set"? — Metaphysician Undercover

Most definitely. The empty set is a set. No set is no set.

Perhaps it's a bit clearer now? — Metaphysician Undercover

Yes. You are confusing the axiom of extension, which tells us when two sets are the same, with the other axioms that give us various ways to build sets or prove that various sets exist.
Taking from the infinite.
I don't see that as a trivial point, — Metaphysician Undercover

What's trivial is saying that the Vitali set is "specified" because all its elements are real numbers. That's like saying the guests at a particular hotel this weekend are specified because they're all human. It's perfectly true, but it tells you nothing about the guests at the hotel. That's why your point is trivial.

because not only is "set" undefined, but also "element" is undefined. — Metaphysician Undercover

If $x \in y$ is true, then If $x$ is an element of If $y$ . That's perfectly well defined in terms of If $\in$ , which is an undefined primitive. I've referred you many times to the axioms of Zermelo-Fraenkel set theory, or ZF, which if you'd read the page, might answer many of your questions.

It's also sometimes called ZFC in honor of Mrs. Zermelo, who was pro choice.

So we have a vicious circle which makes it impossible to understand what type of thing a set is supposed to be, — Metaphysician Undercover

On the one hand you're the only one who doesn't understand this. On the other hand, Skolem argued that the notion of set is too vague to be useful as a foundation for mathematics, and he was one of the greatest of the early set theorists. So you're not wrong. It would be better if you had more sophisticated arguments, because it would then be more interesting and fun to engage with you.

and what type of thing an element is supposed to be. — Metaphysician Undercover

If $x \in y$ then the thing on the left is an element of the set on the right. In pure set theory the thing on the left is also a set; and in set theories with urelements, the thing on the left might not be. Likewise in applications the thing on the left might be something else such as a voter in social choice theory, or a rational actor in an economic theory, etc.

What is a set? — Metaphysician Undercover

The axioms don't tell us. A set is characterized by its behavior under the axioms. Even you've agreed to that previously.

It's something composed of elements. What is an element? It's a set. — Metaphysician Undercover

So what? You want an explicit definition, but in set theory there are no such definitions. Read the axioms. You're just recapitulating Frege's complaints to Hilbert, but I can't argue the point because there's no right or wrong to the matter. You don't like it, that's your right.

Under this description, a particular set is identified by its elements, not by a specification, definition, or description. Do you see what I mean? — Metaphysician Undercover

LOL. Yes, that's the axiom of extensionality, which I've been explaining to you for at least two years. Glad you finally got it. A set is entirely characterized by its elements.

Under your description, any particular set cannot be identified by the predicates which are assigned to the elements, because it is not required that there be any assigned predicates.. — Metaphysician Undercover

Some sets are specified by predicates, such as the set of all natural numbers that are prime. Some sets aren't specified by predicates, such as the Vitali set.

And as I've noted, constructivist mathematicians, neo-intuitionists, finitists, and ultrafinitists only believe in sets that can be constructed by an algorithm or explicit procedure. So this is a fair debate in the philosophy of math.

But there still might be such an identified set. — Metaphysician Undercover

Like the set of natural numbers that are prime. That's a set given by the axiom schema of specification.

So a set must be identified by reference to its members. T — Metaphysician Undercover

"By reference?" No. The Vitali set is characterized by its members, but I can't explicitly refer to them because I don't know what they are. It's a little like knowing that there are a billion people in China, even though I don't know them all by name.

his is why, under this description of sets, the empty set is logically incoherent. A proposed empty set has no members, and therefore cannot be identified. — Metaphysician Undercover

On the contrary. Since everything is equal to itself, the empty set is defined as $\{x : x \neq x\}$ . I rather thought you'd appreciate that, since you like the law of identity. The empty set is in fact the extension of a particular predicate.

If, on the other hand, a set is identified by it's specification, definition, or description, (which you deny that it is), then there could be a definition, specification, or predication which nothing matches, and therefore an empty set. — Metaphysician Undercover

Exactly. The empty set is the extension of the predicate $x \neq x$ . Or if you like, it's the extension of the predicate "x is a purple flying elephant." Amounts to the same thing.

Hopefully you can see that the two, identifying a set by its elements, and identifying a set by its predications, are incompatible, because one allows for an empty set, and the other does not. — Metaphysician Undercover

Since the empty set is the extension of a particular predicate, your point is incoorect.

So as much as "set" may have no formal definition, we cannot confuse or conflate these two distinct ways of using "set" without the probability of creating logical incoherency. — Metaphysician Undercover

You're wrong, since the empty set is the extension of a predicate.

By saying that "set" has no definition, we might be saying that there is nothing logically prior to "set", that we cannot place the thing referred to by the word into a category. — Metaphysician Undercover

In set theory that's true. Although in the formal theory, the axioms and the inference rules of first-order predicate logic are taken as logically prior.

But if you make a designation like "there is an empty set", then this use places sets into a particular category. — Metaphysician Undercover

Well it's in the category of sets. Which is formally true in the category of sets, and is also true in the sense of "category" that you're using. A set is a set.

And if you say that a set might have no specification, this use places sets into an opposing category. If you use both, you have logical incoherency. — Metaphysician Undercover

I don't see why. Every set is fully characterized by its elements. Specification is one particular way of identifying or constructing or proving the existence of particular sets. Other ways are union, intersection, and so forth. It's all given in the axioms, which I've given you the link to many times and which I'm sure you've never even bothered to glance at. You know the most credible way to make an argument is to take the trouble to become familiar with the thing you're arguing against. You won't do that. So you make very trivial and sophistic arguments.

Therefore it is quite clear to me, that the question of whether a set is identified by reference to its elements, or identified by reference to its specification, is a non-trivial matter because we cannot use "set" to refer to both these types of things without logical incoherency. — Metaphysician Undercover

I don't see why. Now do I agree that constructive mathematicians do make this sharp distinction between sets that can be constructed by an algorithm or mechanical procedure (there are many sub-flavors of this idea) versus mathematicians who believe in nonconstructive sets such as the Vitali set. @sime, for example, is one who makes this distinction. If you wish to make an argument along these lines, that would be an interesting conversation. But I don't think that's what you're doing. I don't know what you're doing. i don't know what your point is.

tl;dr: There are many sophisticated arguments against set theory. Therefore you're not shocking me by questioning set theory. But your arguments are from ignorance rather than knowledge. So your arguments are not interesting.

ps -- Am I being to harsh to your ideas? I can't really follow your logic. The axioms are very clear. There's no circularity or infinite regress involved. There's no formal definition of set in the axioms; but some sets are given by specifications and others not. They're not "defined" by the specifications, rather their existence is given by the the axiom of specification. Some other sets are given by various other axioms. Replacement is one of the more subtle ones. Perhaps it's helpful to think of the axioms as a toolkit for determining which sets can exist. In that respect, specification isn't all that special. As to what it all means, perhaps it means nothing at all. That's not a problem, neither does chess.
Taking from the infinite.
Absent AC, it is undecided whether there is such a set. — TonesInDeepFreeze

Yes ok I meant assuming the negation. But if you don't specify one or the other, you're right.
Taking from the infinite.
They are equivalent in Z, so, a fortiori, they are equivalent in ZF. But they are not logically equivalent. — TonesInDeepFreeze

Is Z supposed to be Zermelo set theory? In Wiki they call it $Z^-$ , is that the same thing?

https://en.wikipedia.org/wiki/Zermelo_set_theory

Why do you work in Z so much, why do you care, why exactly should I care? I'm not familiar with it. Perhaps you can supply some context please.

Z |- AC <-> ZL & ZL <-> WO & AC <-> WO

But it is not the case that

|- AC <-> ZL & ZL <-> WO & AC <-> WO — TonesInDeepFreeze

Why is that last line not true? What exactly do you mean? Are you just saying it's not a tautology? Can you give a model in which it's false? In my experience, the claim that "AC, WO, and ZL are equivalent" is totally noncontroversial and nobody (but you) would look twice at it. Why do you take exception?

What I'm looking for is context and understanding of your thought process.
Taking from the infinite.
OED: specify, "to name or mention". Clearly the set you called "V" is not unspecified, and it's you who wants to change the meaning "specify" to suit your (undisclosed) purpose. Sorry fishfry, but you appear to be just making stuff up now, to avoid the issues. — Metaphysician Undercover

Whatever, man.

"suit your (undisclosed) purpose" -- what does that mean? I'm using specification as in the axiom schema of specification. We are talking about set theory after all. If we're talking about baseball, a "fly" is a ball that's hit by the batter and remains in the air without touching the ground. It's not a winged insect.

What "undisclosed purpose" would I have? The corruption of the youth? What are you talking about?

Nevermind, I don't want to know.

If as you agree, all sets in standard set theory are composed of nothing but other sets; and that therefore every nonempty set whatsoever can be said to have elements that are sets; then isn't the fact that the elements of any set have in common the fact that they are sets, a rather trivial point? Can you see that this is not a helpful criterion to specify which elements are in the set and which are not? It's the example I gave of the doorman at a highly exclusive club who's told to just let everyone in. To be a valid specification, you have to tell the doorman which people to let in, and which to not let in.

But like I said, I am perfectly willing to accept your personal definition of the word; at the cost of no longer being able to take you even slightly seriously; since you're making such an unserious point.

AC, ZL and WO are not logically equivalent. But they are equivalent in Z set theory. — TonesInDeepFreeze

Not sure I follow. You mention Z a lot but that's a pretty obscure system unless one is a specialist. AC, ZL, and WO are surely equivalent in ZF. In what way would you say they're not equivalent?
Taking from the infinite.
if one works with a normed linear space that is separable, the Hahn-Banach theorem doesn't require it either. — jgill

Nice to have an actual mathematician around here!
Taking from the infinite.
That's right, to specify that they are real numbers is to specify, just like to specify that the guests at the hotel are human beings is to specify. The fact that a specification is vague, incomplete, or imperfect does not negate the fact that it is a specification. — Metaphysician Undercover

Only if you change what a specification is. In set theory, a specification is a predicate, a statement that can be true or false of a given item. The items for which the predicate is true, go into some set.

By saying, "Oh, they're all real numbers" when the set in question only contains SOME of the real numbers, does not specify the set.

But your definition of specification is not even valid by the everyday English meaning of the word. If you go into a store to buy a computer and you ask the salesman for the specifications of a particular model you have in mind, and they say, "Oh, it's a material object made of atoms," you'd walk out of the store. He has told you the truth, but has not given the specifications of the computer.

Likewise if you are a manufacturer and you subcontract out a particular part, asking to have it made to a particular specification, and they send you back a lump of metal totally unlike what you asked for, saying, "Well it's a physical object made of atoms, what more to you want," you'd sue them for breach of contract.

I told you how so. You've specified that the set contains real numbers. You are the one who explained to me, that 'set" is logically prior to "number", and that not all sets have numbers as elements. This means that "set" is the more general term. How can you now deny that to indicate that a particular set consists of some real numbers, is not an act of specifying? — Metaphysician Undercover

You have not specified which real numbers are in the set and which aren't. And if you really believed what you are saying, you would have told me long ago that every set is either empty or contains other sets; and THAT is a specification. Is that what you claim? Every set is fully specified by saying its elements are other sets? That's nonsense. That's not what a set specification is.

A set specification consists of two things: One, an existing set; and two, a predicate saying which elements of the given set are members of your specified set. You haven't done that. You've made a trivial and sophistic point, saying that every set of real numbers is "specified" because it contains real numbers. That's nonsense. It's childish.

Good, you now accept that every set has a specification. — Metaphysician Undercover

Only by completely changing both the mathematical AND the everyday meaning of specification.

Do you also agree now that this type of specification, which "doesn't tell me how to distinguish members of a set from non members", is simply a bad form of specification? — Metaphysician Undercover

Abraham Lincoln used to ask, If you call a tail a leg, how many legs does a dog have? And he answered: Four. Calling a tail a leg doesn't make it a leg.

Likewise, calling your vague and sophistic characterization of a set a specification does not make it a specification. Of course it could be argued that logically IF I call a tail a leg, then a dog has five legs. In THAT sense, you have made your point. Which is to say: You haven't made your point.

Do you now see, and agree, that since a set must be specified in some way, then the elements must be "the same" in some way, according to that specification, therefore it's really not true to say that "the elements of a set need not be "the same" in any meaningful way." — Metaphysician Undercover

Not in the least. Of course in pure set theory (ie set theory without urelements), the elements of every nonempty set are other sets. So it's true that the elements of every set are sets, and they have that in common. But that is not a property that distinguishes any given set from all other sets, so it's not a specification. If you want to call it a specification that's your right, just as you can call a tail a leg and say that a dog has five legs. If you can get anyone to take you seriously.

So we can get rid of that appearance of contradiction by stating the truth, that the elements of a set must be the same in some meaningful way. To randomly name objects is not to list the members of a set, because a set requires a specification. — Metaphysician Undercover

You haven't specified a set by pointing out that all its elements are sets. Because that doesn't distinguish that set from any other set. But like I say, you can hang on to your childish sophistry or you can hang on to your credibility. You might as well hang on to the former, having long ago lost the latter.

What I am trying to get at, is the nature of a "set" You say that there is no definition of "set", but it has meaning given by usage. Now I see inconsistency in your usage, so I want to find out what you really think a set is. Consider the following. — Metaphysician Undercover

Again -- AGAIN -- you are equivocating between the definition of a set in set theory, of which there is none -- just check the axioms please -- and the fact that some sets are given by specifications, where a specification consists of an already-existing set and a predicate. See the axiom schema of specification for an explication of this point. It's called a schema because it actually consists of infinitely many axioms, one for each predicate.

Since we now see that a set must have a specification, — Metaphysician Undercover

The Vitali set has no specification. It's true that all its elements are real numbers; and in fact all its elements are also sets, since in set theory, real numbers are modeled as sets. But that's as I said a childish and sophistic point, which you can hold to only at the loss off your own credibility.

do you see how the above quote is inconsistent with that principle? — Metaphysician Undercover

No, I see you making an astonishingly childish and sophistic point, calling a tail a leg and saying a dog has five legs.

Since a set must have a specification, a set is itself an "articulable category or class of thought". — Metaphysician Undercover

Most sets have no specifications. You've lost all credibility with me at this point. Well to be fair, you lost all credibility last week when you denied that pi is a particular real number. I have no idea what you might be thinking, but whatever it is, it's wrong.

And, it is not the "being gathered into a set" which constitutes the relations they have with one another, it is the specification itself, which constitutes the relations. — Metaphysician Undercover

Sure, trivially. But not meaningfully, since it's circular. How do I tell which real numbers are in the Vitali set? Well, they're in the Vitali set. Not helpful.

So if you specify a set containing the number five, the tuna sandwich you had for lunch, and the Mormon tabernacle choir, this specification constitutes relations between these things. That's what putting them into a set does, it constructs such relations. — Metaphysician Undercover

If it makes you happy to hold to this line of argument, I would not take that away from you.

Now here's the difficult part. — Metaphysician Undercover

Oh this should be good.

Do you agree that there are two distinct types of sets, one type in which the specification is based in real, observed similarities, a set which is based on description, and another type of set which is based in imaginary specifications, a set produced as a creative act? — Metaphysician Undercover

Only to the extent that your first category is empty. You ask if there's a set based on "observed" similarities. I have never observed a set. I've been to math grad school and never observed a set. Set's are entirely abstract objects and are not observable in the sense of physics nor in the sense of everyday English. You cannot observe a set.

And "real?" What on earth do you mean by that?

FWIW I will grant that you might have meant to ask: Are there some sets given by the axiom of specification; and others that are not, and that are essentially nonconstructive? In which case yes, the set of even natural numbers is an example of the former, and the Vitali set the latter. But neither set is particular "real" or "observable." I have never observed an even number. I've seen four apples. And after they downgraded poor old Pluto, I then knew about the eight planets. But four? Or eight? I've never observed them. They're both abstract entities.

Do you acknowledge that these two types of sets are fundamentally different? — Metaphysician Undercover

Sure. All the sets that there are, are in the latter category; and none are in the former. No sets are based on anything "real" or "observed," and all sets are "based in imaginary specifications, a set produced as a creative act." Those are the only types of sets there are.

Just as there are two types of elephants: those that fly, and those that don't fly. That is a true statement. It's just that all the elephants are in the latter category and none in the former.

But being charitable and assuming you meant to ask about constructive and nonconstructive sets, sure. Constructivist mathematicians make the distinction all the time. Just ask @sime, who rejects the axiom of choice. He does not believe in the Vitali set (I assume, though we've never directly discussed it), and believes all sets are the output of some algorithm or deterministic process. It's a fair distinction.
Arguments for livable minimum wage.
...Why would you need price controls? Supply would just go up do meet demand and so bring prices back down. Or it wouldn't (because people can conspire to limit supply), but then the same would happen with water.

Basically, what's different between houses (where you predict a rise in prices will lead to shortages) and water (where you predict a rise in prices will will met by a rise in supply and so even out)? — Isaac

Excellent point. In the abstract, no difference. In practice, huge difference. Water is liquid (no pun intended); housing isn't.

In a water shortage, price controls prevent "gouging" so nobody has to feel they're being treated unfairly. But nobody has an incentive to truck in water from the next county. So the price of water remains low, but nobody can get any.

If the price is allowed to rise to meet demand, the newspapers will complain of "gouging," but entrepreneurs will truck in water to take advantage of the potential for profits. Prices go up but supply matches demand and everyone who can afford it gets a drink. I suppose there can be government subsidies for those who can't afford the new prices. Hard to know what to do about these cases.

Housing is highly illiquid. If you give everyone more money and impose rent control, rents will rise and many people will be unhoused. If you let prices float upward, it won't help much because you can't truck in new housing. It takes years to get new housing developments approved, especially in cities where the problem is the most acute. And counterintuitvely, new housing is often opposed by low-growth advocates of the leftist persuasion, and upscale liberals who want more housing for the poor, just not near them. We see this in real life all the time. Cities impose rent control and then make it difficult or impossible to build new housing, resulting in massive housing shortages.

The problem (in our country, anyway), is not that people can't afford rent, it's that their inability to afford it is covered by the state. Since it's not in the state's best interests to just let people go unhoused (it needs a ready-to-work workforce in 'reserve' to accommodate economic growth), it has to pay landlords where the unemployed and low wage earners can't afford to. The landlords know this and so set the rent accordingly. — Isaac

Not in the US. Except in the past year, where the government is freezing rents and bailing out landlords. Making an absolute sucker out of anyone who scrimped and saved to honorably pay their rent. That's a big problem with bailouts, moral hazard. It makes a fool out of anyone who actually paid their debts. It incents people to be deadbeats.

Prices are not set in a vacuum. If I have figs to sell in the market, I don't pick a price point at random and then see how they go. I pick a price point using my knowledge of the world. I know figs are quite common, I've bought them myself in the past etc. In a world where there was a minimum wage system in place, that would be one of the bits of information about the world I would use to set my prices. If I put my prices so high that some people can't afford them, the RPI would go up, minimum wage would go up, corporation tax and wages would go up to cover it, and I'd end up making a loss. — Isaac

Housing isn't figs, as I've noted. Historically, free markets do better than controlled economies in terms of vibrance, growth, doing business, and establishing the right price. Controlled economies lead to misallocation of resources.

To be fair I'm now arguing an Austrian economics position about which I know the buzzwords but not the details. I should probably quit while I'm behind here because I'm already in a bit over my head. But controlled economies like Cuba and Venezuela don't do well. The USSR collapsed. China grew when they introduced some strange kind of state-controlled capitalism. I'm not entirely sure how their system works.

All minimum wage is is a system for ensuring that there's no economic gain to be had from a corporation pricing it's essential goods beyond that which it's lowest paid workers can afford. — Isaac

It has been argued that minimum wage laws cause unemployment. If a low-skill worker costs more than a robot, that worker can't get a job. You see this all the time in the news. I support minimum wage laws because without them it would be a race to the bottom and people would be forced to work for pennies. We don't want a society like that. But if you set the rate too high, especially in a world with a huge surplus of unskilled labor, you create massive unemployment. Which you solve with welfare programs, again making a sucker of anyone who works for a living.

If they do, the system simply corrects the wage to meet it so no increase in net profit is possible that way. Profits have to be made on luxury items instead. — Isaac

I'm sure you remember the famous story of the luxury tax on yachts passed 30 or so years ago by the US Congress. The idea was to soak the rich. What happened instead was that boatyards went out of business, causing unemployment among their blue-collar, working-class employees.

WAPO has the story, wan't hard to find, I just googled, "luxury tax yachts" and the story popped right up.

https://www.washingtonpost.com/archive/business/1993/07/16/how-to-sink-an-industry-and-not-soak-the-rich/08ea5310-4a4b-4674-ab88-fad8c42cf55b/

As a result, in its first year and a half, the yacht tax raised a pathetic $12,655,000 for the Treasury. That's enough to run the Agriculture Department for a little over two hours. Meanwhile, the tax has contributed to the general devastation of the American boating industry -- as well as the jewelers, furriers and private-plane manufacturers that were also targets of the excise tax that was part of the 1990 budget deal.

But Senate Majority Leader George J. Mitchell (D-Maine), Sen. John H. Chafee (R-R.I.), Sen. John Breaux (D-La.) and Rep. Benjamin L. Cardin (D-Md.), all of whom coincidentally represent boating states, are sailing to the rescue, and repeal of the luxury tax is included in both the House and Senate versions of the budget reconciliation bill.
Taking from the infinite.
So are you agreeing that mathematical infinity has neither philosophical nor scientific relevance — sime

Not entirely, but I'm not disagreeing either. Infinitary set theory has been used in the 20th century as the foundation of math, and math is the language of physics. So if we say that infinitary math is unrealistic or fiction or nonsense or false (any one of which I'd agree with for sake of discussion), we still have to explain the "unreasonable effectiveness" of math. So infinitary math is of great philosophical importance. And surely there are a whole lot of professional philosophers of math who do discuss and care about infinitary math. So infinitary math is both philosophically and scientifically relevant, even if we can plausibly argue that it's fiction. After all, as I'm fond of pointing out, Moby Dick is a work of fiction, but it still teaches us to avoid following our obsessions to our doom. Even fiction can be useful i the real world.

and that everyone knows this, — sime

I don't think everyone sees this issue the same way. There are a lot of different opinions, even different learned opinions.

or am i right to stand on a soap box — sime

That's always a personal choice. I may dislike the designated hitter rule in baseball, but I don't go out and rant about it in public. One chooses one's battles. There are those who dislike contemporary mask mandates, but wear their mask as required by local laws. There are others who go into grocery stores and confront the hapless clerks. Each of us has many opinions, but we still have to choose which hills to die on and which soapboxes to stand on.

and point out the idiocies and misunderstandings that ZFC seems to encourage? — sime

I disagree with you about this. The way the knight moves in chess is a fiction, it's an arbitrary rule of a formal game. Does it produce idiocies and misunderstandings? No, it's just a rule of the game. I've seen prominent set theorists admit that they don't know if set theory is true or meaningful. Set theory is the study of certain formal structures. People do it because they find it interesting. Others are Platonists and believe they're seeking some higher truth.

I'm not sure what idiocies and misunderstandings you mean. If you don't like the way the knight moves, don't play the game. Or play some alternate variant of chess. There are many alternate variants of set theory. And billions of people live perfectly happy lives without ever knowing or caring about set theory.

So you're not "wrong," pe se, in disliking or objecting to infinitary math. But if in addition to that you have a strong emotional aversion to it, that's ... well, it's a personal issue. You might introspect as to why. Maybe you had a screechy math teacher in third grade. A lot of math anger started that way.

Obviously, a denial of AC doesn't amount to an assertion of ~AC, given that things are generally undecidable, — sime

Ok. Still, absent choice, infinite sets are badly behaved. There's a vector space without a basis, a surjection without a right inverse (or section), a commutative ring with unity with no maximal ideal. These things are very inconvenient in math.

And that's the biggest reason for adopting AC versus rejecting it. Convenience. The reasons for adopting or rejecting axioms are pragmatic. We are not asserting any kind of absolute truth. We're only choosing the axioms that make it convenient to do math. Maddy explains all this in her classic articles, Believing the Axioms, I and II. We want expansive rather than restrictive axioms, and so forth.

but i see no counter-intuitive examples in what you present. — sime

If you don't find a vector space without a basis, a surjection without a right inverse, an infinite set that changes cardinality when you remove one element, etc., counterintuitive, then we see that differently. In the end, AC makes infinite sets well-behaved. For example without AC there are the Alephs, and then there are many infinite cardinalities that aren't Alephs. With AC, all the cardinals are Alephs. There's no "absolute truth" in that, just convenience.

I think if you regard AC as a pragmatic choice, it's easier to understand. It relieves you of needing to stand on a soapbox. Mathematicians are choosing convenience and a more orderly and expansive set-theoretic universe. What need is there to stand on a soapbox against someone's pragmatic choices?

In fact, many examples you raise should be constructively intuitive if we recall that construction can proceed either bottom-up from the assumptions of elements into equivalence classes, or vice versa, so an inability to locate a basis in a vector space using top-down construction seems reasonable.p/quote]

If you're making a constructivist argument, I can't argue with you. Many people agree with your point of view.
— sime
As for the sciences, AC is meaningless and inapplicable when it comes to the propositional content. At best, AC serves a crude notation for referring to undefined sets of unbounded size, but ZFC is a terribly crude means of doing this, because it only recognises completely defined sets and completely undefined sets without any shade of grey in the middle as is required to represent potential infinity. — sime

I suppose I can concede your point that AC is not strictly necessary for the sciences. Nor is the knight move necessary to cook a lasagna.

QM has also been reinterpreted in toposes and monoidal categories in which all non-constructive physics propositions have been removed, which demonstrates that non-constructive analysis is dying and going to be rapidly replaced by constructive analysis, to the consternation of inappropriately trained mathematicians who resent not knowing constructive analysis. — sime

ZFC is not even a 100 years old. In its present form it dates from Zermelo's 1922 axiomatization. And Mrs. Zermelo was pro choice. (/joke). There's no telling how these matters will be seen in another hundred years. As Max Planck said, science proceeds one funeral at a time. Meaning that the old guard die off and a new generation grows up accepting the new ideas.

Obviously, the axiom of choice isn't used in the finite case. In the infinite case, the sets of states needs to be declared as being Kuratowski infinite in order to say that the elements of the set are never completely defined, and so a forteriori the size of the set cannot be defined in terms of it's finite subsets. — sime

Why can't a nation in some alternate universe have infinitely many states, and choose a legislature? My argument here is that although AC is independent of ZF, it's nevertheless intuitively true. Even the ld joke admits that. AC is definitely true, the well-ordering theorem is definitely false, and Zorn's lemma, who knows! The joke being that they're all logically equivalent.

Secondly, the set should be declared as Dedekind finite, in order to say that the set is an observable collection of elements and not a function (because only functions can be dedekind-infinite). — sime

You lost me there. Absent AC there is a set that is infinite (not bijective with any natural number) yet Dedekind-finite (no proper subset is bijective with the entire set). I don't know what you mean here.

So, yes, you can choose as many representatives as you wish without implying a nonsensical completed collection of legislatures that are a proper subset of themselves, but formalisation of these sets isn't possible in ZFC, because AC and it's weaker cousin, the axiom of countable choice, forces equivalence of Kuratowski finiteness and Dedekind finiteness. — sime

I don't think you're engaging with my point. Why can't the uncountably many provinces of the planet Zork choose themselves a legislature? Why on earth can't we form a set consistent of one element from each of a collection of nonempty sets? It's intuitively true that we can, even if AC is not provable from ZF.

Besides, Gödel showed that AC is true in his constructible universe in which, as I understand it, all ordinary mathematics takes place anyway. So there is a perfectly good model of math in which AC is true. You can't stand on a soapbox and deny that.

But if you prefer to do math without AC, or if you are a constructivist, or a finitist or ultrafinitist, you're in good company. I can't argue you out of your preferences. I can only question your soapbox emotions. After all most serious constructive mathematicians still use some version of AC, because without it it's more difficult to do math.
Why the Many Worlds Interpretation only applies to a mathematical universe.
If you read the wiki article you linked, you'll see that MWI is a level 3 classification scheme of multiverse theory. I was correct then. — Philosophim

They seem to be using multiverse in a more general way, so in that sense you're right. My understanding of multiverse is as in eternal inflation, which is not a QM theory. I'd agree that there's a considerable confusion between the multiverse and the MWI.

You seem to have side stepped the larger issue I made however. — Philosophim

I didn't sidestep it, I didn't address it at all.

In the end, MWI is a unicorn theory. Do you have an answer for this? — Philosophim

I have no answer for that. Multiverses as in bubble universes are plausible, in the sense that there are parts of the universe that are causally unreachable from the observable universe. We can't know what's out there. MWI I don't find very plausible, but I don't know enough QM to have an informed opinion.
Taking from the infinite.
I still don't see your point, or the relevance.
— Metaphysician Undercover

The point is that basing your mathematical "principles" on empiricism or reality demonstrably leads to absurdity, including your rejection of fractions, negative numbers, imaginary numbers, infinity, circles, probabilities, possible set orderings, and potentially all mathematics. Instead of coming to realise that this indicates a serious problem with your principles and position, you continue in your delusion that you possess a superior understanding of mathematics. — Luke

:100: :100: :100: :100: :100: :100: :100: :100: :100: :100: :100:
Meno's Paradox
In summary, Meno's paradox assumes that when someone says, "I don't know," this person has no proposition that could initiate an inquiry, a dubious assumption. — TheMadFool

Falsified by Galileo, who asked questions about things he didn't know, and did experiments to find out. For example he rolled balls down an inclined plane to discover gravitational acceleration. That's how science works.

Look at it this way. There are things you don't know and don't know enough about to even frame a question.

Then there are things you don't know, but still conceptualize well enough to frame a question. This is the domain of science.

In the words of the great American philosopher and lousy secretary of Defense Donald Rumsfeld (recently deceased, Rest in Hell), there are the known unknowns and the unknown unknowns.

Meno is talking about the unknown unknowns, the subjects of which our ignorance is so profound that we can't even frame a question. But he's forgetting about the known unknowns: the subjects about which we are ignorant, yet about which we are able to frame questions and conduct experiments. That's the realm of science.

From Galvani making frog legs twitch when connected to an electric spark, to the modern digital computer. That's the march of the known unknowns. Meno soundly falsified.
Why the Many Worlds Interpretation only applies to a mathematical universe.
Could be that there's actually zero energy in the universe. — Michael

Way above my pay grade. If Sean Carroll didn't make a video about it, I have no idea :-)
Why the Many Worlds Interpretation only applies to a mathematical universe.
Mind explaining how? — Philosophim

MWI is as you explain it, branching due to QM, as an alternative to wave function collapse.

The multiverse theory says that the universe consists of "bubble universes" that branch off and are causally independent of each other. Entirely different theory. Nothing to do with quantum branching. It's a cosmological theory.

https://en.wikipedia.org/wiki/Multiverse
Why the Many Worlds Interpretation only applies to a mathematical universe.
Multiverse theory is the same as unicorn theory — Philosophim

Could well be. But for purposes of this discussion, please note that multiverse theory and the many-worlds interpretation are two entirely different speculative theories.
Taking from the infinite.
You are specifying "the real numbers". How is this not a specification? — Metaphysician Undercover

The real numbers include some numbers that are in $V$ and many that aren't. In what way does that specify $V$ ? That's like saying I can specify the people registered at a hotel this weekend as the human race. Of course everyone at the hotel is human, but humanity includes many people who are not registered at the hotel.

Actually, you're wrong, your set is clearly a specified set. — Metaphysician Undercover

How so? I gave an existence proof. In no way did I tell you how to determine which real numbers are in it and which aren't. Can you explain your thought process?

This is not true, you have already said something else about the set, the elements are real numbers. — Metaphysician Undercover

And the people at the hotel are humans. As are all the people not at the hotel. If that's all you mean by specification, that all I have to do is name some arbitrary superset of the set in question, then every set has a specification. If that's what you meant, I'll grant you your point. But it doesn't seem too helpful. It doesn't tell me how to distinguish members of a set from non members.

It's like an exclusive club that only allows in certain people. You run the club. You hire a doorman. He asks, "How can I tell who's a member or not?" And you say, "Oh just let everybody in." What kind of specification is that?

I'll agree with Tones, the two ways are just different ways of looking at the same thing. That's why I said the Wikipedia article is consistent with the SEP. I do believe there are metaphysical consequences though, which result from the different ways, or perhaps they are not consequences, but the metaphysical cause of the difference in ways. The principal consequence, or cause (whichever it may be), is the way that we view the ontological status of contingency. — Metaphysician Undercover

I agree that there is a point of view by which it makes no difference whether you define infinite regress as going forward or backward. And a sense in which it makes a huge difference, such as well-foundedness. It can be argued either way.
Taking from the infinite.
Footnote 1 of the SEP article says: "Talk of ‘first’ and ‘last’ members here is just a matter of convention. We could just as well have said that an infinite regress is a series of appropriately related elements with a last member but no first member, where each element relies upon or is generated from the previous in some sense. What direction we see the regress going in does not signify anything important." — TonesInDeepFreeze

Awfully good catch, thank you. Especially since the footnotes don't appear on the article page and must be clicked on to see them at all. I commend your attention to detail in clicking on the footnotes.

I am of two minds on this. On the one hand yes, the footnote is correct and there is fundamentally no difference mathematically. The two interpretations are order anti-isomorphic, just mirror images of each other.

Still, to me the semantics are profoundly different. I guess I have to accept that in the end the difference is not important. Still @Metaphysician Undercover must also agree that when he says that @jgill and I have infinite regress wrong, he's incorrect about that too. If both interpretations are the same, everyone's right.

Thanks for clicking on that footnote!

ps -- I just can't agree with SEP, period. The article is wrong and I'm right. My basis for this belief is the concept of well-foundedness, which is essential to set theory and is encoded as the axiom of foundation.

In set theory it is legal to have infinitely upward membership chains $x_0 \in x_1 \in x_2 \in \dots$

In fact this is not only legal, it's standard, as exemplified by the finite von Neuman ordinals $0 \in 1 \in 2 \in \dots$ .

Whereas it is expressly forbidden to have infinitely downward membership chains $\dots \in x_2 \in x_1 \in x_0$ .

Even though the two conditions are mirror-images of each other, set theorists strongly distinguish between the two; considering one situation normal and the other illegal. I rest my case, and will probably drop the author a note with my two cents.
Taking from the infinite.
I won't continue — Metaphysician Undercover

Ah, the good old daze. That didn't last long.

Another example of the division between mathematics and philosophy. But the Wikipedia entry is consistent with the SEP.. You two just seem to twist around the concept, to portray infinite regress as a process that has an end, but without a start, when in reality the infinite regress is a logical process with a start, without an end. — Metaphysician Undercover

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical. — Metaphysician Undercover

It's the distinction between two linear orders:

... < a4 < a3 < a2 < a1 < a0.

There's no first element but there is a last element. The earth (a1) rests on the back of turtle a1, which rests on the back of turtle a2, and so forth. It's turtles all the way down.

The SEP article reverses this:

a0 < a1 < a2 < a3 < ...

The earth (a0) has a turtle on it, a1; which has a turtle on its back, a2, and so forth. It's turtles all the way UP!

Now I recognize that in some sense these structures are "the same," in the sense that they just a mirror image of each other. Technically we would say that there is an order anti-isomorphism between them.

But the semantics are completely different. In the first model, there is no uncaused cause. William Lane Craig would argue (sophistically, but whatever) that this is impossible; that there MUST be a first cause, which is not only God, but is the Christian God. That's the argument.

So my contention is that the SEP article flips the direction of what an infinite regress is. And that's not necessarily a mathematical view. It's the philosophical view too. Turtles all the way down versus turtles all the way up.

This is what I've argued is incoherent, the assumption of an unspecified set, and you've done nothing to justify your claim that such a thing is coherent. I will not ask you to show me an unspecified set, because that would require that you specify it, making such a thing impossible for you. So I'll ask you in another way. — Metaphysician Undercover

I will gladly show you an unspecified set, one of the classic cases. It's called the Vitali set. Consider a binary relation $\sim$ defined on pairs of real numbers by:

$x \sim y$ if $x - y \in \mathbb Q$

That is, two real numbers $x$ and $y$ are related by $\sim$ if their difference is rational. For example $\pi + \frac{1}{2} \sim \pi$ . That's because their difference, 1/2, is rational.

You can verify that the relation $\sim$ is reflexive (every real number is related to itself, since the difference with itself is zero, which is rational); symmetric: if $x \sim y$ then $y \sim x$ ; since the difference in one direction is just the negative of the difference in the other, so they're either both rational or neither are; and transitive, meaning that if $x \sim y$ and $y \sim z$ then $x \sim z$ . You should see if you can convince yourself that this is true.

A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation. There is a basic theorem about equivalence relations, which is that they partition a given set into a collection of pairwise disjoint sets whose union is the original set.

So $\sim$ partitions the real numbers into a collection of pairwise disjoint subsets, called equivalence classes, such that every real number is in exactly one subset. By the axiom of choice there exists a set, generally called $V$ in honor of Giuseppe Vitali, who discovered it, such that $V$ contains exactly one member, or representative, of each equivalence class.

You can tell me NOTHING about the elements of $V$ . Given a particular real number like 1/2 or pi, you can't tell me whether that number is in $V$ or not. The ONLY thing you know for sure is that if 1/2 is in $V$ , then no other rational number can be in $V$ . Other than that, you know nothing about the elements of $V$ , nor do those elements have anything at all in common, other than their membership in $V$ .

Is this example important? Yes, it's part of the foundation of modern probability theory.

$V$ is the classic example of a nonconstructive set. I don't expect you to regard this as particularly intuitive. It's an example typically shown to first-year grad students in math. It takes a while to get your mind around it. But "whether you like it or not," as Gavin Newsom said about gay marriage, $V$ is a perfectly legitimate set in ZFC, and actually turns out to be of theoretical importance.

You have now seen the classic example of a nonconstructive set.

We agree that a set is an imaginary thing. But I think that to imagine something requires it do be specified in some way. — Metaphysician Undercover

You're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.

I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions.

That's the point I made with the distinction between the symbol, and the imaginary thing represented or 'specified' by the symbol. The symbol, or in the most basic form, an image, is a necessary requirement for an imaginary thing. Even within one's own mind, there is an image or symbol which is required as a representation of any imaginary thing. The thing imagined is known to be something other than the symbol which represents it. So, how do you propose that an imaginary thing (like a set), can exist without having a symbol which represents it, thereby specifying it in some way? Even to say "there are sets which are unspecified" is to specify them as the sets which are unspecified. Then what would support the designation of unspecified "sets" in plural? if all such sets are specified as "the unspecified", what distinguishes one from another as distinct sets? Haven't you actually just designated one set as "the unspecified sets"? — Metaphysician Undercover

What distinguishes one set from another as distinct sets? Their elements, as expressed by the axiom of extensionality, as I've explained to you at least a dozen times in the past year.

It's like being out in a field picking daisies. You pick this daisy, you pick that daisy. When you're done, you have a basket full of daisies. Must they have some particular property in common for you to have picked them? No, you picked them randomly. The only thing they have in common is that you picked them. For no reason at all. It's just like this week's winning lottery numbers. They have nothing in common other than that they were picked randomly. Once you start thinking about it that way, you'll find many such examples in daily life of perfectly random sets. A bunch of people check into a hotel.What do they have in common that distinguishes them from all other human beings? Nothing at all, except that they all checked into the hotel.

If you know nothing else about mathematical sets, know this: A set is entirely characterized by its elements.

As Judge Reinhold said to Sean Penn in Fast Times at Ridgemont High: Learn it. Know it. Live it.
Why the Many Worlds Interpretation only applies to a mathematical universe.
This confronts a pretty pernicious issue as to what or which kind of wavefunction collapses cause this to occur along with the extent of the parent universe splitting to what localized or even global effect(s)? — Shawn

I'm already in way over my head and know nothing of this other than a couple of Sean Carroll videos on Youtube. My understanding is that MWI avoids wavefunction collapse. The wavefunction doesn't collapse; rather, everything happens. The cat is awake and asleep, as Carroll says. No reason to kill a cat. He mentions that Schrödinger's daughter said, "My father just didn't like cats."
Why the Many Worlds Interpretation only applies to a mathematical universe.
However, if one assumes in physics, as do many physicists, that the world is not mathematical, then doesn't it mean that conservation of energy laws would become violated for every branching of wavefunction collapses? — Shawn

Sean Carroll explains this by saying that the energy splits too. Each world takes with it half the energy of the parent world, so that conservation of energy is preserved. Of course that means that if the total energy of the universe is finite, at some point there's not enough energy to split into any more worlds. I'd like to ask Sean Carroll that. I'm sure he's thought about it.
Arguments for livable minimum wage.
...Which is it? — Isaac

Two separate cases. If people have more money for rent but rents are capped then there's no incentive to provide more housing. If water is scarce and the price of water ISN'T capped then people have an incentive to provide more water. You compared the price-controlled case to the non-price-controlled case.

In the real world we always see rent control accompanied by severe housing shortages. If you artificially cap the market price of housing, landlords don't build more of it.
Arguments for livable minimum wage.
It kind of feels like minimum wage should be made livable that a person working 40 hours a week should require no public assistance — TiredThinker

Look at it this way. If you give every renter in your town an extra $1000 a month, with no corresponding increase in the number of available rental units, they'll just bid up the cost of rents and eventually absorb the $1000. In other words you get inflation, which is simply an increase in the money supply with no corresponding increase in the availability of goods, resulting in higher prices.

The only way to make such a system work long term, is to give everyone free money and simultaneously implement rigid price controls. And then you create shortages. If everyone has an extra $1000/month for rent and there is no increase in available housing and there is no increase in rents, the available units will quickly be filled to 100% capacity and there will be no place to live despite the extra money in your pocket.

A variant of that argument applies to "price gouging" during emergencies. If the price of water goes way up, people are incented to supply more water. The price is higher, but everyone can get the water they need. If you artificially cap the price, then nobody gets "gouged," but many people can't get water; because there is no incentive for anyone to supply more water.

Likewise congestion pricing for services like Uber. If there's a lot of demand, prices go up. Prices go up so more drivers decide to go out and work, providing the market with more supply. If you outlaw congestion pricing, everyone pays the same but you can't find a ride, because you've removed the incentive for drivers to go out and work instead of staying home.
Driving the automobile is a violation of civic duty.
No man needs to drive. It is a violation of civic duty. — Sha'aniah

For sake of conversation, would you have made the same argument in the days of the horse and buggy? Do you object to the "Surrey with a fringe on top?"
Are you an object of the universe?
Man, it seems to me, gives itself a special status among existing things — Daniel

As do cats. That's a cat joke. But in fact every sentient creature does the same. Cats, dogs, the more intelligent insects. I lived in a rural area once and had to kill many living creatures, which I always felt bad about. You can't get so caught up in the beauty of it all that you let the wildlife run free in the house. One thing I learned is that everyone likes to eat and everyone likes to live. Each sentient creature is the center of its own universe. Man is hardly unique in that respect.
Taking from the infinite.
I'm not reverting back. Just because I understand better what I didn't understand as well before, doesn't mean that I am now bound to accept the principles which I now better understand. — Metaphysician Undercover

You seemed to be reverting back to the Frege-Hilbert paradigm, which is a pointless discussion because there is no right or wrong, just a different worldview. I can't talk you out of yours nor would I if I could.

I suggest you look into the concept of infinite regress. The negative numbers are not an example of infinite regress. — Metaphysician Undercover

In fact they are. They are often used philosophically as a model of infinite regress of causation. If you say cause -47 causes -46 which causes -45 etc., you have a model of causality in which (1) every effect has a direct cause, yet (2) there is no first cause. That's infinite regress.

ps -- I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. Going forward without end like the Peano axioms is not an infinite regress. I refer to all the standard cosmological arguments, for example William Lane Craig's Kalam cosmological argument, where he argues against infinite regress going backwards. I have never seen infinite regress defined incorrectly as going forward as in this SEP article. The author made a mistake.

You might check the Wiki article on infinite regress, which is itself a little vague but at least correct. The SEP piece confuses induction with infinite regress. That's false. Induction always has a base case. Infinite regress fails to have a base case, that's what makes it an infinite regress.

No, you said "set" has no definition, as a general term, and I went along with that. But I spent a long time explaining to you how a set must have some sort of definition to exist as a set. — Metaphysician Undercover

You are equivocating two senses of "definition." The word set has no definition in set theory. You can consult the axioms of Zermelo-Fraenkel set theory to verify this.

On the other hand, some sets do have definitions, or more accurately, specifications. For example the set of prime numbers, the set of even numbers, the set of counterexamples to Fermat's last theorem. That latter by the way is the empty set. See the axiom schema of specification to understand how SOME sets may indeed have a specification.

But MOST sets can't possibly have specifications, because there are more sets than specifications, a point I've made several times and that you prefer not to engage with. There are uncountably many sets and only countably many specifications. There simply aren't enough specifications to specify all the sets that there are. Most sets are simply collections of elements unrelated by any articulable property other than being collected into that set.

And a set can't ONLY be given by a specification, because then I'll give you the specification $x \notin x$ and get the Russell paradox.

It's a bit like numbers. There is no general definition of number; but there are specific definitions of the real numbers, the complex numbers, the p-adic numbers, the quaternions, and so forth.

You seem to be ignoring what I wrote. — Metaphysician Undercover

I carefully read everything you write. And either refute it or place it in context or give my own point of view. And then you come back with the same misunderstandings and ignore my refutations.

Since you haven't seriously addressed the points I made, — Metaphysician Undercover

I have addressed all of your points and either refuted them or placed them in context.

and you claim not to be interested, — Metaphysician Undercover

I'm not interested in recapitulating the Frege-Hilbert dispute, since it's a matter of worldview and I could not change your mind because there is no right or wrong about the matter. And I'm in good company, because in the end Hilbert simply stopped responding to Frege's letters. If you reject abstract axiomatic systems, nobody can talk you out of that viewpoint.

I won't continue. — Metaphysician Undercover

Thanks for the chat, then. All the best.

Thanks for sharing your wisdom on these types of threads — Gregory

You're welcome. Very much appreciated.

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