The problem of pedestrian travel versus logistical transport is to address the separation of church and state. No man needs to drive. It is a violation of civic duty. The community needs to be partitioned out according to these boundaries. There should be no driving in the town square. Colleges were developed to resemble small towns, and in the first two years on college campus one is not permitted to have a car. The town needs to be restored and people need to embrace walking as a neglected medium between men. The automobile is a carbon-emitting, steel-pod isolate. Isolate, here, is a noun. It disallows spontaneity in economics, such as walking in off the street. It separates people and the exchange of visibility is off-kilt. I get it frequently, "I saw you out walking." But I did not see you..? This is a problem in the social fabric. It is a conscientious problem, and a humanitarian one. If a person lives in one town, then I expect them to work, sleep, and fellowship there too, as well as shop. Now, excessive mobility has made it so that I work in town A, sleep at town B, and have friends and shopping in town C. That is not good social welfare, and it creates frailed, untrusted relationships. Also it puts wal mart in business instead of more small businesses. — Sha'aniah

I wish you'd stop holding back and tell us how you REALLY feel about cars.

And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. — sime

I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo.

Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians. — sime

May well be so. I still think the way the knight moves is nonsensical too. What of it? You don't find me down at the park yelling at the chess players. Why is this a concern to you?

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem.

Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot.

So it seems to me a number is a "unity" — Gregory

I always get into trouble with these philosophically loaded terms. Any number can be broken up into parts. 2 = 1 + 1, 1 = 1/2 + 1/2. So nothing in math is "indivisible." If anything at all is, it would be a pure Euclidean point, or a single real number representing a location on the number line. But what of it? Making a mystery or a big deal out of the idea that something is a "unity" doesn't speak to me; and it's one of the points where I do get in trouble in these philosophical discussions. A number is a number. It might be a real number, representing a signed distance on the number line. Or it could be a complex number, representing a rotation and stretching operator in the plane. Or a quaternion, used by game programmers as a nice formalism for rotating things in 3-space. I know the formalism of how numbers are represented in set theory; and I know that numbers aren't "really" sets; rather, they're abstract things that are pointed to by their various representations. What that means, I don't worry about too much. I've read a bit of the literature, I was reading up on structuralism the other day when that came up in one of the discussions on this site.

But I don't know what a unity is, or whether a number is one. I looked it up on Google, and it says that, "Unity, or oneness, is generally regarded as the attribute of a thing whereby it is undivided in itself and yet divided from others."

Well ok. But other than 0 and an individual point in space, I don't know what it means for any mathematical entity to be "undivided in itself." Actually only a point on a line has no parts. A point in space, say 3-space, is given by three spacial coordinates (x,y,z). And that's three things! I can take a point's projection onto the x, y, and z axes, to find that its "components" are x, y, and z, respectively. So even a point in space has components. I don't know what philosophers make of that.

All it all I can't agree or disagree that a number is a unity. I don't even know what that means.

and a set is not a noun but more like a verb. It's our action of containing a unity or many unities or unities and containers (verbs). — Gregory

I see what you mean. You have an apple and an orange, and forming a set {apple, orange} is an act of gathering. It's quite mysterious and not entirely coherent, a point @Metaphysician Undercover has made and that I somewhat agree with. I don't know what it means to form a set out of individual objects. When pressed I can fall back on the formalism of the axiom of pairing, one consequence of which is the fact that if I have a mathematical object $X$, I'm allowed to form $\{X\}$, "the set containing" $X$. I totally understand and accept that mathematically. Metaphysically, I don't know for sure that it's even a coherent concept.

That in fact is one of @Metaphysician Undercover's frequent points. What he doesn't understand is that I totally agree with him. Or at least I do for sake of discussion. It's not a hill I need to die on. I make no claim that set theory is coherent or sensible. Only that it's a formal system of rules that some people find interesting. I don't reify set theory or put it on a pedestal or make any claims about it. Like the novel Moby Dick. It's not a true story, but it's worthwhile nevertheless. It's based on a true incident, but only very loosely. If someone wants to tell me that set theory is incoherent, I don't object to that point of view. It doesn't matter. It's interesting on its own terms; and massively useful in formalizing most of modern math. What more can you ask of a formalism?

Which is to say that if set-collection is regarded by you as a verb, I do see your point. The act of gathering individuals into a mathematical set is a great act of abstraction that leads to many counterintuitive results. It's a powerful concept, even if not entirely coherent.

I've been considering the "set of all sets that do not contain themselves" vs the "set of all sets the do contain themselves". — Gregory

Russell's paradox just shows that we can't form the set of all sets; and in fact that we can't form sets out of arbitrary predicates. Now that's very profound. Originally it was thought that if P is a predicate, then the collection of all the things that satisfy P form a set. That turns out to lead to a contradiction. Rather, a set is nothing more or less than exactly what the axioms of set theory say they are. Which for some philosophers is not a very satisfactory state of affairs. What I do know is that "high school sets," which are collections of similar or related objects, are nothing like actual mathematical sets. Actual mathematical sets are far stranger than that.

This leads to what I see as Hilbert's position (contra Frege) of our rational power of humans to think of thinking of thinking of thinking and on to infinity. The set\verb would take precedence over the unity\number we place before our eyes as an object. — Gregory

I was trained in modern mathematical abstraction and have a hard time understanding Frege's point of view. It upsets some people (Frega, @Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. As Hilbert said, "“One must be able to say at all times
— instead of points, straight lines, and planes — tables, chairs, and beer mugs.” Whether he truly believed that, or was only retreating behind the formalist view because a realist mathematical stance is untenable, I don't know. Hilbert's formalist dreams were blown up by Gödel. There's a realm of mathematical truth that exists outside of anything we can capture with axioms.

Sorry. Some phrases allude me. I read into them too much.

Just making a little joke, didn't mean for it to get so involved!. But definitely appreciated learning that it was Samuel Johnson's witty remark. Now I don't feel so bad having made such a lame joke :-)
— TiredThinker

https://www.theidioms.com/a/ — TiredThinker

Thanks for that link.

That's the world that MU lives in. — jgill

:100:

I thought we were heading for a China hawk manifesto about the need to confront China militarily. — T Clark

Pay no attention to those million Uyghurs behind the curtain.

Would you have said the same in 1935 about u-no-hoo? Asking for six million friends.

Again, this is the difference between fiction and fact. — Metaphysician Undercover

But we're not talking "fact," if by that you mean the real world. The subject was set theory, which is an artificial formal theory. Set theory is not any part of any physical theory. I pointed out to you that in set theory, everything is a set, including the elements of sets. You responded by saying you hadn't realized that. I thought we were therefore making progress: You acknowledged learning something you hadn't known before. And now you want to revert back to "fact," as if set theory has an ontological burden. It does not.

We can imagine infinite regress, and imagine time extending forever backward, but it isn't consistent with the empirical evidence. — Metaphysician Undercover

But I never claimed it did. I offered the mathematical example of the integers. Are you a disbeliever in sufficiently small negative numbers? Do you believe in -47? -48? -4545434543? Where does your belief stop? Of course this is not a physical example, it's a mathematical example; in fact, an example that illustrates the difference between physics and math.

That's the problem with infinite regress, it's logically possible, — Metaphysician Undercover

Ok! Then we are in agreement. Since I have made absolutely no other claims. So just to satisfy my curiosity, do you believe in the negative integers? They believe in you.

but proven through inductive (empirical) principles (Aristotle's cosmological argument for example) to be impossible. — Metaphysician Undercover

Discussion for another time, but I have made no claims about the world. Why do you argue as if I did?

I beg to differ. Didn't we go through this already in the Gabriel's horn thread. — Metaphysician Undercover

That was a lengthy thread from a while ago. Can you remind me of the specifics? It's not possible that "we went through this" about the Riemann sphere. Stereographic projection is a commonplace idea among every mapmaker since antiquity who's wrestled with the dilemma of representing a spherical earth on a flat map. Can you remind me of what on earth you might be talking about? You place a sphere above a plane. From the north pole of the sphere, you draw a straight line through a point on the sphere and extend the line to a point on the plane. You thereby have a mapping from the sphere to the plane. In cartography it's a basic technique. In complex variables theory, it's a way of visualizing the complex numbers as a sphere. There is no mysticism or "vicious circle" or any such nonsense as you claim.

It seems like you haven't learned much about the way that I view these issues. — Metaphysician Undercover

If you would say what you're talking about, I can respond. The Gabriel's horn thread was lengthy and long past. Tell me what you're talking about.

In any event, I've learned far too much about how you view things.

You write very well, but your thinking hasn't obtained to that level. Another example of the difference between form and content. — Metaphysician Undercover

Your ignorance is only matched by your ill manners. Going forward, if you can't be civil, put a sock in it.

Are you denying the contradiction in what you wrote? — Metaphysician Undercover

I repeatedly said that the only thing they have in common is being elements of the given set. So why are you acting like I haven't said that every single time?

If they are members of the same set, then there is a meaningful similarity between them. — Metaphysician Undercover

Only in a sophistic sense. I already pointed out to you that if "meaningful similarity" or "property" or "predicate" is interpreted as referring to an idea expressible in a finite-length string of symbols, there are more subsets of the natural numbers than there are properties. Therefore most sets are entirely random. Their elements have nothing at all in common except for being gathered into the given set.

Being members of the same set constitutes a meaningful similarity. — Metaphysician Undercover

Ok fine, on that definition. I'll agree. But it's a pretty trivial point. Especially for you to be going on about it.

You said "the elements of a set need not be 'the same' in any meaningful way. — Metaphysician Undercover

Other than being in the same set. You deliberately quote me out of context to make a point. Disingenuous much?

The only thing they have in common is that they're elements of a given set." Can't you see the contradiction? — Metaphysician Undercover

No. What I said is perfectly accurate.

If they are said to be members of the same set, then they are the same in some meaningful way. — Metaphysician Undercover

Only that they are members of the same set. So what? You are being childish to go on like this.

It is contradictory to say that they are members of the same set, and also say that they are not the same in any meaningful way. — Metaphysician Undercover

Yeah yeah.

Another example of this same sort of contradiction is when people refer to a difference which doesn't make a difference. If you apprehend it as a difference, and speak about it as a difference, then clearly it has made a difference to you. Likewise, if you see two things as elements of the same set, then clearly you have apprehended that they are the same in some meaningful way. To apprehend them as members of the same set, yet deny that they are the same in a meaningful way, is nothing but self-deception. Your supposed set is not a set at all. You are just saying that there is such a set, when there really is no such set. You are just naming elements and saying "those are elements of the same set" when there is no such set, just some named elements. Without defining, or at least naming the set, which they are members of, there is no such set. And, naming the set which they are elements of is a designation of meaningful sameness. — Metaphysician Undercover

Why are you going on like this? Let me remind you of the conversation. You expressed realization that in set theory, everything's a set. Then you claimed that leads to infinite regress. I pointed out that one, there's nothing logically wrong with infinite regress. I gave the negative integers as an example.

Then I pointed out that in set theory, we adopt the axiom of foundation to explicitly rule out infinite regress. You totally ignored both those points to go off on this trivial and pointless tangent.

Here is a feature of imaginary things which you ought to learn to recognize. I discussed it briefly with Luke in the other thread. An imaginary thing (and I think you'll agree with me that sets are imaginary things, or "pure abstraction" in your terms) requires a representation, or symbol , to be acknowledged. And, for an imaginary thing, to exist requires being acknowledged. However, the symbol, or representation, is not the imaginary thing. The imaginary thing is something other than the symbols which represent it. So the imaginary thing necessarily has two distinct aspects, the representation, and the thing itself, the former is called form, the latter, content. And this is necessary of all imaginary things. — Metaphysician Undercover

Well, for sake of discussion, it's not clear to me that every imaginary thing has a referent. Sets, for example. The empty set is an imaginary formal thing, but I don't know that it has a referent. Certainly not in the physical world.

The important point is that you cannot claim to remove one of these, from the imaginary thing, because both are necessary. So a purely formal system, or pure content of thought, are both impossibilities. And when you say "these things are elements of the same set", you have in a sense named that set, as the set which these things are elements of, thereby creating a meaningful similarity between them. The point being that a meaningful similarity is something which might be created, solely by the mind and that is how the imagination works in the process of creating fictions. But when something is a creation, it must be treated as a creation. — Metaphysician Undercover

Focus. Focus. You said that the fact that in set theory everything is a set, leads to infinite regress. I pointed out that the negative integers are an example of an unproblematic negative regress; and that the axiom of foundation rules out infinite regresses of set membership.

You have avoided both those points to go off on trivialities and irrelevancies. And personal insults. What's the point?

Again, incoherency fishfry. Can't you see that? There is necessarily a reason why you place them in the same set, and this 'reason why' is something other than actually being in the same set. — Metaphysician Undercover

Oh no, not at all. The powerset of the natural numbers is uncountable. There are more sets than reasons. Most sets have no reason at all.

You've gone from saying that the elements have something in common, namely being in the given set -- which I agree with -- to now saying that there's some OTHER reason in addition to that. You're simply wrong about that. The powerset of the natural numbers exists, that's an axiom of set theory. Every set has a powerset, the set consisting of all the set's subsets. And the powerset is far larger than the set itself. There aren't enough "reasons" or predicates or explanations to cover them all, by a countability argument.

You are not acknowledging that "being gathered into a set" requires a cause, — Metaphysician Undercover

You're thinking of the south and the Civil war. A side in a war needs a cause. A set needs no cause. Show me in the axioms for set theory where it says that. This is just something you made up. Again, you're trying to reify sets; but sets are only imaginary formal entities whose behavior is entirely determined by the axioms.

and that cause is something other than being in the same set. — Metaphysician Undercover

You're just making that up. And changing the subject.

I challenged you on your claim that the idea that sets contain only other sets leads to infinite regress. I pointed out that the axiom of foundation precludes infinite regress of set membership. You changed the subject.

So the relation that the things have to one another by being in the same set is not the same as the relation they have to one another by being caused to be in the same set. — Metaphysician Undercover

You can say the knight flies over the moon, but that's not in the rules of chess. There are no "causes" in the axioms of set theory. So you're just making this up and then typing in crap, and wasting my time trying to get you to focus on the actual conversation we were having, which for a brief moment got substantive before you reverted to just making things up.

And things which are in the same set necessarily have relations to each other which are other than being in the same set, because they have relations through the cause, which caused them to be in the same set. — Metaphysician Undercover

There are not enough predicates to cover all the sets that there are. Most sets have no reason or cause at all; they're pure randomness.

It appears like you didn't read what I said. — Metaphysician Undercover

I could say the same about you. But I have read what you've said. What you've said is wrong; and your repeating it doesn't make it any less wrong.

That a word is not defined does not mean that it has no meaning. As I said, it may derive meaning from its use. If the word is used, then it has meaning. So if "set" derives it's meaning from the axioms, then there is meaning which inheres within, according to its use in the axioms. — Metaphysician Undercover

Ok. Fine. But there are no "causes" in the axioms.

What we do not agree on is what "inherent order" means. — Metaphysician Undercover

Don't start that crap again. I can't help it if you reject modern math. I can't do anything about that.

i really do not see how you get from the premise, that "set" is not defined, but gets its meaning from its use, to the conclusion that a set might have no inherent order. In order for the word "set" to exist, it must have been used. Therefore it is impossible for "set" not to have meaning, and we might say that there is meaning (order, if order is analogous to meaning, as you seem to think), which inheres within. Wouldn't you agree with this, concerning the use of any word? If the word has been used, there is meaning which inheres within, as given by that use. And, for a word to have any existence it must have been used. — Metaphysician Undercover

You've worn me out. I'm losing interest.

It appears like you misunderstood. I didn't say every set is a number, to the contrary. I said that if we proceed under the precepts of set theory, every number is a set. — Metaphysician Undercover

Well as Bill Clinton said, that depends on what the meaning of "is" is. If you mean that a number literally is a set, no, that's not true, as Benacerraf so insightfully pointed out. If you mean that in set theory a number is represented by a set, then that's true. Important for you to make that distinction.

Therefore we cannot say that "number" is undefined because "set" is now a defining feature of "number", just like when we say every human beings is an animal, "animal" becomes a defining feature of "human being". — Metaphysician Undercover

In set theory, a number is defined as a particular type of set. Just because set is an undefined term doesn't mean that we can't use it to define other things. Just as point is an undefined term in Euclidean geometry, but a line is made of points. Right? Right.

Didn't it strike you that I was in a very agreeable mood that day? — Metaphysician Undercover

Yes, that didn't last long. But you were more than agreeable the other day. You actually achieved some insight. You realized that a set has no definition, and that its meaning is derived from the axioms. You realized that the members of sets are also sets.

Now I'm back to my old self, pointing out your contradiction in saying that things could be in the same set without having any meaningful relation to each other, other than being in the same set. You just do not seem to understand that things don't just magically get into the same set. There is a reason why they are in the same set. — Metaphysician Undercover

You're just wrong about that. Provably wrong, since there aren't enough reasons to cover uncountably many sets.

Maybe at some point we'll discuss the supposed empty set. How do you suppose that nothing could get into a set? — Metaphysician Undercover

By the axiom of pairing, which has as a consequence the fact that if $X$ is a set, so is $\{X\}$. Everything's given by the axioms.

I have no idea what you mean by "nothing." That's not in the axioms. The empty set is not nothing. It's the empty set. A particular thing.

Actually I do not agree with general relativity, so I would ban that first. — Metaphysician Undercover

Charming. You don't believe in abstract math, you don't believe in physics.

You keep saying things like this, the Pythagorean theorem is not true, now Euclidian geometry in general is not true. I suppose pi is not true for you either? Until you provide some evidence or at least an argument, these are just baseless assertions. — Metaphysician Undercover

You probably shouldn't bring up pi. You said the other day that pi is not a particular real number. That's a statement so monumentally ignorant that I either have to ignore it or stop responding to you altogether. So far I'm just trying to ignore it. Why you'd bring it up again, I don't know. You're just reminding me what a monumental waste of time this is.

On what basis do you say they are a unity then? [/url}

The axiom of powersets.
— Metaphysician Undercover

You have a random group of natural numbers. Saying that they are a unity does not make them a unity. — Metaphysician Undercover

Every subset of the natural numbers is a set.

So saying that they are a "set" does not make them a unity. This is where you need a definition of "set" which would make a set a unity. — Metaphysician Undercover

I'm afraid "unity" is not mentioned in the axioms. You keep making things up. You are unable to focus on what's in the axioms. It's like someone trying to teach you chess and you say, "Well the knight must wear armor and save damsels," or "The knight must be "a man who served his sovereign or lord as a mounted soldier in armor." No no no no no. The knight in chess is exactly what the rules say the knight is. You don't get the concept of formal rules, fine. I doubt you're like this in real life, and you're quite tedious to regress to this infantile obfuscatory state here. I thought we'd moved a little past that, but apparently not.

Then you have no basis to your claim that a set is a unity. — Metaphysician Undercover

But I never said a set is a unity. I don't know what a unity is. It's not mentioned in the axioms.

And you cannot treat a set as a unified whole. If a set is supposed to be a unified whole, then you cannot claim that "set" is not defined. — Metaphysician Undercover

I agree that objection has been raised against set theory. It's not a point I'm interested in debating. Thoralf Skolem pointed out that the concept of set is far less coherent than people imagine. Many mathematicians and philosophers have made the point. For purposes of discussion, I'll even concede the point. But it's irrelevant. It doesn't diminish or change set theory, which is a particular formal system that need not have any referent or even be entirely sensible. It's just a list of formal symbols and the game is to derive their logical consequences.

If you don't want to play chess, that's fine. But for you to stand on a soapbox in the middle of town and rant and rail about chess, that's another thing entirely. You don't like set theory, you get no argument from me. I like set theory but I don't think others need to. But your vociferous objections to the reality of set theory are a waste of time. I don't make any claims it's real. It's just a formal system that some people find interesting, and that gained 20th century mindshare as the foundation of math. In fact set theory is all the more interesting lately, "now that it's been relieved of its ontological burden," as one set theorist put it.

ps -- @Meta let me sum this up. A couple of weeks ago I noticed that you are taking Frege's side in the great Frege-Hilbert controversy; namely, that you claim axioms must mean something or refer to something. Hilbert says no, that the theorems must be true of beer mugs and tables.

Since you feel that way, it's not something I can talk you out of. There is no right or wrong position. In real life Frege refused to "get" modern math and Hilbert stopped returning his letters. Likewise you don't want to get modern math. That's your right. But there is no point in your repeating these same talking points. The axioms of set theory are what they are. There are no "causes" or "reasons" nor "inherent order." I can't argue these points with you anymore.

We made a bit of progress when you started to at least acknowledge the reality of modern set theory. But if you don't want to build on that, I can't argue you out of your position and I wouldn't if I could.

pps --

I do reject fractions, — Metaphysician Undercover

LOL.

Here we can see clearly the dichotomy, so if it were unclear before it should be very much clearer now. In our day to day life we have light switches and power generation as separate entities. — kudos

Programmers know that distinction as interface versus implementation. It's not a particularly deep idea. If you swapped out a coal-fired power plant for a nuclear one, the operation of the light switch would not change even though the underlying implementation is completely different.

In the mind we have it organized that way as well. — kudos

Well yes, civilization is composed of layers of abstractions.

Our subjective relation to technological means conditions us to believe in things that do and things that make do. — kudos

It's unavoidable. You can't master auto mechanics to drive a car, and power generation to turn on the light. You are stating everyday commonplaces as if they held some kind of deep insight.

Shouldn’t it make sense that we think of Mathematics in the same light? — kudos

Sure, and we do. We take theorems as given without necessarily caring about the centuries of hard work it took to develop the insight to prove the theorem. It's human progress. You don't have to invent concrete to lay down a highway.

After all, we all use Matlab/Octave/etc. Nobody wants to compute a giant integral that will take all day. — kudos

You seem to be confusing the computation aspects of mathematics with actual mathematics. Possibly you're not overly familiar with the latter.

This type of reasoning is tempting but can be fallacious, for the reasons previously explained. — kudos

Yes but you're the only one committing the fallacy. You talk of Matlab is if it were a stand-in for actual mathematics. And it's not clear what fallacy you are talking about. It's not a fallacy to use a light switch. It's just an example of a user interface, just as a web browser frontends the entire global communications infrastructure of the Internet.

The concepts of mathematics are most commonly acknowledged as valid through proof; proof that heavily involves the form of computation. — kudos

I'm guessing that you haven't seen much math, because once again you conflate mathematics with computation. Some proofs involve computation, but most don't.

We can only create once we have seen for ourselves that the dualism was never wholly and fully mutually exclusive. — kudos

Yeah ok. Whatever point you are trying to make is deeply unclear, and muddled by your lack of specific experience with mathematics, as opposed to computation.

If you had never heard of power generation perhaps the best way to prove it to you might be to use the switch, at least as an aide as opposed to persuading you by recourse to theories of electron — kudos

But the operation of a light switch proves nothing to anybody about power generation. The entire purpose of a light switch is to relieve the end user of the burden of even thinking about power generation.

interactions that haven’t been observed and haven’t been synthetically proven from prior knowledge. Those theories are like light switches to the subject of what that switch means to us as human beings.
15 minutes ago — kudos

Buzzwords and word salad. You are saying nothing. Your exposition is devoid of meaning. Feel free to convince me otherwise.

You seem to be making a big deal out of the fact that there are interfaces and implementation. Which is fine, if trivial. But your attempt to connect the idea to mathematics falls flat, since you think mathematics is computation. And you haven't made any point about it in any case.

ps -- I'm not giving you a hard time just to do that. I can not understand what you are saying, and the parts that I do understand, are wrong. I'm challenging you to be more clear.

The conclusion of the above considerations, then, is that driving the automobile in the above context is a violation of civic duty. — Sha'aniah

If you don't like the way I drive, stay off of the sidewalk!

Well imagine a perfect programming language so easy to use every citizen could create any program they wanted no matter how complex by simple computations without having to know much about programming. — kudos

Like COBOL, "Common business-oriented language," hyped in the 1960's as a way to let business people write their own programs without the need for professional programmers?

What would be the long term effects of having these types of programs? — kudos

The fantasy would fail, just as it did for COBOL, graphical programming, and every other "non-programmer" programming paradigm ever hyped. You could do your homework and write an article on the history of failed approaches to the idea of programming without programmers.

Would you say it would promote a deeper experiential understanding of the mechanics and interrelationships within those functions not to have any experiential interaction with them any more? Certainly it could, but do you think it would? — kudos

Not only don't I think it would, but we have six decades of actual real-world experience that the idea doesn't even work. It turns out that you need programmers to write programs. COBOL became a success only because professional programmers used it. Business people never did.

But how would higher-level tools to let nonspecialists write programs enable a "deeper experiential understanding?" Does driving an automatic transmission give you a deeper experiential understanding of how transmissions work? Does flipping a light switch give a deeper experiential understanding of power generation and distribution? Of course not. The higher-level the interface, the less actual understanding is involved.

The entire purpose of high-level abstractions is to relieve the end user from the burden of understanding what's going on under the hood. If you want someone to understand how software works, they should program in assembly, not high-level languages. You go down the stack, not up, in order to understand what's going on. You go up the stack to get things done without the need to know what's going on.

None of this his anything whatsoever to do with your OP, which seemed to be about the distinction or dichotomy between programming and mathematics. Here you're talking about methods of letting non-programmers write programs, an idea with six decades of abject failure behind it. [I'll concede spreadsheets as the one known success. Maybe simple SQL queries executed by business people, though the organization still needs to employ an army of database administrators]. You didn't explain your OP at all.

To commit to polarization would make the concept less and less real, as its computation became easier and easier it would require less and less intervention of mind. — kudos

If you could give a specific example, your post would be more clear. As it is, I can't figure out what you're saying.

This remark is reported to have been said by Samuel Johnson in James Boswell's Life of Samuel Johnson, 1791. The occasion was Johnson's hearing of a man who had remarried soon after the death of a wife to whom he had been unhappily married. — TiredThinker

You looked it up rather than tried to figure it out? You ARE a tired thinker! But thanks for the reference, I did not realize the remark originated with Johnson.

Based on this it almost sounds like the first marriage didn't last long enough to have mattered? — TiredThinker

His EXPERIENCE with marriage was not happy; nevertheless he HOPED the next marriage would be better. Hence the triumph of hope over experience.

The more people you sleep with and love the more numb you become and the less you truly value someone — MAYAEL

Don't gourmets eat a lot of great food, and thereby hone their palates? Don't lovers of great music attend many concerts, and thereby increase their appreciation of musical excellence?

↪fishfry what does that mean? — TiredThinker

If you had to guess, what would you say?

Is there something special about first marriages? Like that first person will always be a bigger deal than any spouse after? — TiredThinker

You know what they say about second marriages. "The triumph of hope over experience."

pluralistic monism — Enrique

Is that like a square circle?

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself.
— fishfry

This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. — Metaphysician Undercover

There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random.

This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object. — Metaphysician Undercover

I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact.

Suppose you arbitrarily name a number of items and designate it as a set. — Metaphysician Undercover

Ok.

You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. — Metaphysician Undercover

Tru dat. Or as the kids say, Yes, indubitably so.

After all as I just noted, there are only countably many criteria, formulas, computer programs. But there are uncountably many sets of natural numbers. Most of those sets are entirely random. There is no rhyme or reason to their constituent members. They're just random collections.

Unless you are a constructivist, in which case you deny the existence of random sets. Some go down that path.

In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. — Metaphysician Undercover

Only their collection into a set.

If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance. — Metaphysician Undercover

Do you deny the existence of all sets that cannot be cranked out by a Turing machine or at least defined in first-order logic? You can do that if you like. I don't see the use. Consider the following thought experiment. You flip a fair coin a countably infinite number of times. You thereby generate a sequence of 1's and 0's. What invisible magic forces the resulting bitstring to be computable, or describable by an algorithm or formula? Why can't the result be completely random, having no pattern at all? That's by far the most likely outcome.

You write very well. That must be why I like to engage with you, not that I want to troll you. — Metaphysician Undercover

Thank you. I'll get to your second post later, I'm falling behind.

In set theory everything is a set.
— fishfry

I didn't know that, but it makes the problems which I've apprehended much more understandable. — Metaphysician Undercover

Yes. Everything is a set. Or what they call a "pure set," meaning a set whose elements are also sets. There are as I mentioned set theories with urlements, also called atoms, but these are niche theories and not of interest to us at present. So everything is a set that contains other sets.

If everything is a set, in set theory, then infinite regress is unavoidable. — Metaphysician Undercover

No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system.

However in set theory there is no infinite regress. That's guaranteed by the axiom of foundation, also known as the axiom of regularity. It says that no set is a member of itself and it also rules out all circular membership chains like $a \in b \in c \in a$ and so forth. In standard set theory all sets are well-founded. That means that if you take its elements, which are themselves sets; and take their elements, which are themselves sets; and drill all the way down; you are guaranteed to hit bottom. There is no possible infinite regress of sets.

For completeness I'll mention that people do study non well-founded sets, but this is yet another niche interest and of no interest to us here. In standard set theory all sets are well-founded. There can never be an infinite regress of sets.

A logical circle is sometimes employed, like the one mentioned here ↪jgill to disguise the infinite regress, but such a circle is really a vicious circle. — Metaphysician Undercover

Oh jeez man, you embarrassed yourself a little here. See now I feel bad pointing out that you embarrassed yourself because you complimented me. LOL.

@jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me.

I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. — Metaphysician Undercover

I find this deeply inconsistent with other things you've said. Earlier I was making the point that we can have two sets, X and Y, with a bijection between them, and we can say they are "cardinally equivalent," without knowing what that exact cardinal number is. Then later we can define cardinal numbers, and assign one of them to X and Y.

You claimed that the cardinal numbers were already "out there" waiting to be assigned. You used that idea to claim that I was wrong about ordinals being logically prior to cardinals.

So you somehow manage to believe in the existence of cardinal numbers, which include the endless hierarchy of gigantic cardinals given to us by Cantor's theorem: that a set's powerset is always of a strictly larger cardinality than the set. So we have the cardinality of the natural numbers, which is smaller than the powerset of the natural numbers, which is smaller than the powerset of the powerset of the natural numbers, and on and on forever.

You believe in the metaphysical existence of all of these humongously unimaginable cardinals; yet you deny the existence of the empty set on which they're all founded.

That's logically inconsistent.

But never mind that. I don't believe in the existence of the empty set either. Not in reality. If I see a table with nothing on it, there's nothing on it. I do NOT see the empty set sitting on the table. So I agree with you, I don't believe in the empty set.

But I DO believe in the empty set as a formal construction in the game of math. In fact the empty set is the extension of the predicate $x \neq x$. Surely you must agree with that, since you believe in the law of identity.

Can you clarify your remark? If you don't believe in the metaphysical existence of the empty set, I'm in complete agreement. But if you claim to disbelieve in the empty set as a mathematical object, that's like disbelieving in the way the knight moves in chess. You can't disbelieve in it, it's just one of the rules of the game.

And again; if you are so strong on the law of identity, then you must believe that $\emptyset = \{x : x \neq x \}$.

There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class. — Metaphysician Undercover

I don't know what you mean? What do you mean by "class" in this context? Is 2 in its own class? Why not, it's 1 + 1, right? Although in the past you've denied even that, so I hope we're not going down that road again.

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set.
— fishfry

This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. — Metaphysician Undercover

Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing.

Did you simply not read what I wrote? Do you like to just push my buttons? I say something as clear as day; and you respond by admonishing me that I "ought to recognize" the very thing I've just said. I don't get it. That's why I sometimes think you are trolling me.

Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. — Metaphysician Undercover

Ok, {5, my lunch, the Mormon Tabernacle Choir}. What of it?

That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity. — Metaphysician Undercover

A very disingenuous point. The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set.

This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. — Metaphysician Undercover

Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms.

And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.

But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms.

That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. — Metaphysician Undercover

Completely agreed. And therefore a mathematical set has no inherent order, because that's how sets are used in set theory. Can you see that you've now completely conceded the point?

Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. — Metaphysician Undercover

No question about it. There are philosophers and set theorists who question whether the mathematical conception of set is even coherent. You'll get no disagreement from me on that point. Although just because @Tones forgot to mention that ordinal < means set membership doesn't support your point, it only means @Tones forgot to mention it.

This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread. — Metaphysician Undercover

No question. As Hilbert famously pointed out about the axioms of geometry, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."

That's a perfect expression of the formalist position on axiomatic systems.

There is no set of ordinals, this is the famous Burali-Forti paradox.
— fishfry

This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. — Metaphysician Undercover

Yes, very good! That's essentially the proof. Any set of ordinals is itself an ordinal. Hence there is no set of all ordinals.

So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal. — Metaphysician Undercover

Cesare Burali-Forti couldn't have said it better himself.

I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms. — Metaphysician Undercover

The collection of all ordinals is a proper class. In standard set theory, ZF or ZFC, there are no official proper classes, so "proper class" is a colloquial expression. There are set theories in which proper classes are formalized. Either way, a proper class is a collection that's too big to be a set. The class of all sets, the class of all ordinals, etc.

There is no general definition of number.
— fishfry

This is not really true now, if we accept set theory. — Metaphysician Undercover

I am disappointed that you didn't accept the historical point I made earlier. Zero, negative numbers, irrational numbers, complex numbers, and transfinite numbers didn't used to be accepted as numbers, and now they are. Likewise p-adic and quaternions, two other types of numbers discovered only in the past couple of centuries. "Number" is a historically contingent concept.

If "set" is logically prior to "number", then "set"
is a defining principle of "number". — Metaphysician Undercover

Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number.

That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover

So now you agree that ordinals are logically prior to cardinals? I am glad you have internalized this fact to the point where it now seems obvious to you, when only a few days ago you were strenuously disagreeing.

But so what? It's true that in set theory everything is a set, but that doesn't mean everything is a number. I don't follow your logic.

Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist. — Metaphysician Undercover

I looked that up on Wikipedia and it seemed to be about some kind of mystical emanation from "The One." Lost me, I'm afraid. But I'm shocked that you believe in the vast multitude of gigantic cardinal numbers, while professing disbelief in the empty set.

We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. — Metaphysician Undercover

I don't doubt that you tried to explain this to me and I missed it. Even now I don't think I know the difference between Platonism and realism.

We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true. — Metaphysician Undercover

I'm afraid you lost me a bit there. The number 5 exists as a formal symbol and concept in set theory. What it is "for real" I am not sure. It's the thing that comes after 4, that's the structuralist idea, I think.

If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism. — Metaphysician Undercover

Ok. I admit to being unclear on this. I'm only struck by finding you believing in the pre-existence of the vast array of cardinal numbers, yet disbelieving in the empty sets and set theory in general.

A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist. — Metaphysician Undercover

Ok. Is this a bit structuralist? Natural numbers aren't particular things, but they are the relations among them; that is, 5 is the thing that follows 4, and that's all I need to know about it.

This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not. — Metaphysician Undercover

Ok. But 3's easy. How about the transfinite cardinals? You believe in them yet disbelieve in set theory? That's a hard row to hoe.

But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions. — Metaphysician Undercover

You are now willing to agree with me that there may be some virtue to considering math to be an interesting and useful fiction? @Meta I find you agreeing with my point of view in this post.

I do not mean to argue that dreaming up logical possibilities is a worthless activity. — Metaphysician Undercover

You are mellowing! And agreeing with me!! I must be having an effect. I will say that you have achieved some genuine mathematical insight lately.

What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. — Metaphysician Undercover

So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? We disagree on this. Math is the study of that which is logically possible. Math leaves what's real to the physicists. And of course even the physicists no longer have much interest in what's real, but that's a criticism for another time. But math is not bound by what's real. On this we disagree strongly.

So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination. — Metaphysician Undercover

Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true?

hank you for your commentary. — jgill

You're very welcome, glad that helped.

My initial guess was that a set is something that contains and not something in its own right. — Gregory

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. The empty set, the set containing the empty set, and the set containing the set containing the empty set are three distinct sets. Which we could, if we wanted to, take together into a set! Like this:

$\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}$.

So zero remains a nothingness of anything in that case. — Gregory

Zero is not nothing. Zero is a particular point on the real number line. Or the address or location of a point, if you prefer. Zero is a particular thing. It represents the cardinality of the set of purple flying elephants in my left pocket. Zero is something. Nothing is nothing. Which can be read two different ways!

Very abstract ideas. Couldn't structure just be that which contains a process and thus, like sets which compose it, it is nothing in itself. — Gregory

Well, what's a process? If a process is something that can be "contained," it needs explanation.

This would certainly make mathematics a system of process and divorce it from the notion that anything rests and stays permanent within it — Gregory

The natural numbers seem permanent. They don't change from day to day, whether you regard them as sets, as in the von Neumann finite ordinals, or as a process of starting at 0 and taking successors.

I am no expert on these things though.

Would you even be able to tell the difference? Then being awake would feel no different to being asleep. — Cidat

Nonexistence is a lot different than being asleep. If you've ever had general anesthesia, that's the closest you can get to "experiencing" nonexistence. You are literally not there. Sleep is nothing like that. You dream, you're aware of unusual noises, you toss and turn. Under general anesthesia they turn you off and then turn you back on again when they're done. And believe me, it's a hell of a lot different than being aware of your existence. I don't understand questions like this. When you're awake and aware you experience stuff. When you're dead or nonexistent you don't.

There are articles about the philosophical aspects of general anesthesia, for example

https://pubs.asahq.org/anesthesiology/article/84/5/1269/35549/A-Philosophical-Approach-to-Anaesthesia

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5193047/

https://www.nytimes.com/2013/12/15/magazine/what-anesthesia-can-teach-us-about-consciousness.html

https://mindmatters.ai/2021/02/what-is-your-soul-doing-when-youre-under-anesthesia/

ps -- It occurs to me that during general anesthesia you certainly exist. I think I refuted my own point. Still, anesthesia is a very interesting state of being.

That said, can you say what "this" refers to? Cohen's invention of forcing in general?
— fishfry

That would be good. I had heard the expression but had no idea what it was. The article came as a revelation to me. And here I thought the reals consisted of rationals and irrationals. — jgill

Of course the reals consist of rationals and irrationals. That's provable from the axioms. Every model of the reals satisfies the axioms of the reals. FWIW I'm familiar with the work of Natalie Wolchover, the author of the article you linked. She puts the "pop" in pop science; which is to that that she's very good, up to a point; but not past that point. I didn't read the article and can't vouch for anything she might have said.

I can't really describe forcing. Timothy Chow, the author of A Beginner's Guide to Forcing, describes forcing as an "open exposition problem." That is, just as an open problem is a problem nobody knows how to solve, forcing is a subject that nobody knows how to explain to non-specialists.

I strongly recommend his article for anyone interested in the subject; bearing in mind that nobody would be expected to understand much of it, and the more times you read it and the more you read about forcing in general, the better vague understanding you'll get. But there's no known explanation that's any easier than diving in and learning the actual set theory, and it's a notoriously difficult subject.

The basic idea is that we want to know what things are consistent with a given set of axioms, so we try to find models that satisfy the axioms and also satisfy the extra things we're interested in. For example in geometry we can take the Euclidean axioms minus the parallel postulate (PP). We know the PP is consistent because Euclidean geometry is a model of the axioms plus PP. On the other hand in the 1840's, Riemann and others discovered that the axioms plus not-PP also had a model. This means that you can take the other axioms with PP or with the negation of PP, and both resulting systems are consistent. Alternatively, you can say that PP is "independent" of the other axioms; given the axioms, you can neither prove nor disprove PP.

A slightly more sophisticated example if you've seen group theory is that if you take the axioms for a group, you might want to know whether the "Abelian axiom" is consistent and/or provable; namely, is it true that for all x and y in a group, xy = yx.

Well, the integers with addition are an example of a group in which it's true. But the set of invertible 2x2 matrices with multiplication also form a group, and there are examples where commutativity fails. Since there are models of the group axioms with ah]d without commutativity, we would say that the "Abelian axiom" is independent of the group axioms.

Ok. Now with that in mind, down to cases. Cantor called the cardinality of the natural numbers $\aleph_0$. He proved that the cardinality of the real numbers was $2^{\aleph_0}$. And he showed that the next larger cardinal after $\aleph_0$ is $\aleph_1$.

So we have $\aleph_0$ directly followed by $\aleph_1$. And out there among the Alephs is $2^{\aleph_0}$. The question is, might it be the case that $2^{\aleph_0} = \aleph_1$? This question, or rather the claim that equality holds, is the continuum hypothesis (CH).

Cantor was unable to prove CH, and neither was anyone else. In 1940 Kurt Gödel proved that at the very least, CH was consistent with the other axioms of ZF, Zermelo-Fraenkel set theory. He did this by exhibiting a model in which it was true. This model is called Gödel's constructible universe. It's a universe of sets in which all the axioms of ZF are true, and in which CH is true. This showed that at the very least, assuming CH did not introduce any contradiction into ZF that wasn't already there.

That last remark needs explanation. What do I mean that CH doesn't introduce a contradiction that wasn't already there? Recall that Gödel had already proven in 1931 that ZF can't prove itself consistent. So the only way to know if ZF is consistent is to introduce even stronger principles that in effect assume it is. For all we know, set theory is inconsistent.

What Gödel showed, then, is that if ZF is consistent, so is ZF + CH. That is, all these proofs are relative consistency proofs. They don't show that anything is consistent; they only show that IF one system is consistent, then so is that system plus some other stuff.

But what about the negation of CH? Is that consistent with ZF as well? Gödel had shown that there's a model of ZF + CH. Could there be a model of ZF + not-CH? The problem is that nobody had any idea how to cook up alternative models of ZF. This was a real problem.

In 1963 an analyst named Paul Cohen figured it out. By analyst,I mean he was into real analysis -- epsilons and deltas and convergence and such. About as far away from mathematical logic as you can get. He woke up one day and said to himself, "I think I'll take a run at CH." He figured out how to cook up alternative models of ZF. In 1966 he won the Fields medal, the only Fields medal ever given for mathematical logic. I have always assumed that all the other official professional logicians must have been mighty annoyed. Some nonspecialist wakes up one day and solves the greatest unsolved problem in your field.

So ok all of that is preamble. How did he cook up alternative models? He invented a method called forcing. And having come this far, I really can't say much about it; first, because I don't know much about it myself, and second, because as Timothy Chow noted, nobody knows how to explain this to nonspecialists.

The idea basically is analogous to the procedure in abstract algebra where we adjoin roots to fields. That is, suppose that we believe in the rational numbers. We know the rationals satisfy the field axioms: you can add and multiply rationals to get another rational. Multiplication distributes over addition. And every nonzero rational has a multiplicative inverse.

Now suppose we want to prove that there is a field that contains the rationals and that also contains the square root of 2. We "adjoin" a meaningless symbol, $\sqrt 2$, to the rationals. We know nothing about this symbol other than that it has the formal property $(\sqrt{2})^2 = 2$.

In order to preserve the field axioms we have to say that all possible additions and multiplications are also in our new "extension field," as it's called. So we have a set of expressions of the form $a + b \sqrt 2$. We can then prove that the resulting system of formal expressions $a + b \sqrt 2$ itself satisfies the field axioms.

This is the best analogy for forcing. We start with a model of set theory, and we carefully add new "thingies," whatever they are, making sure that our new system also satisfies the axioms of set theory. If we're clever, we can arrange things so that CH turns out to be false in our new model. Then we collect our Fields medal. Cohen was clever.

So the idea -- and this is pretty much everything I know about it -- is first, we start with a model of ZF. But wait, since we can't prove within set theory that set theory is consistent, we don't know for sure if there even IS a model of set theory. But no worries. If there is no model, then set theory is inconsistent, and then we can prove ANYTHING, including CH and its negation. So, to get things off the ground, we assume that set theory is consistent and that it has a model.

Then -- and this step comes out of nowhere, pretty much -- since there is a model, there is a countable model. This is the famous, or infamous, Löwenheim–Skolem theorem.

What on earth does it mean to have a countable model of set theory? Doesn't ZF prove that there are uncountable sets? Well yes, it does. But now we have to broaden our understanding of what that means. What does it mean for a set to be countable? It means there is a bijection from that set to the natural numbers.

Suppose we have some set X, and a bijection from X to the naturals. So X is countable. Now suppose we have some model of set theory, and we throw out all the bijections between X and the naturals from the model, making sure we still have a model. Then from "inside the model," X would be uncountable; but from outside the model, from our God-like perspective, we can perfectly well see that X is really countable.

So we learn that uncountability is a "relative" notion. A set may be countable in one model, and uncountable in another. It just depends on which bijections are lying around.

So we assume we have a model of ZF; and we then know that if we do, there must be a countable model; and then we can show that if there's a countable model, there is a countable, transitive model.

Having done that, we take a countable, transitive model of ZF, and carefully add sets to it, making sure that we preserve the axioms of ZF, while cooking up a violation of CH.

Well there you go. I wrote a lot of words and didn't explain a thing about what forcing is. Definitely go read Tim Chow's excellent article.

I didn't know about structuralism in math! That the number one is an idea, a true idea, seems to me to be the basis of all that follows though, kinda that unity before the plurality. But structuralism in all forms is a really interesting idea! — Gregory

I found the SEP article interesting. It breaks down all the various sub-genres of mathematical structuralism and talks a lot about whether category theory is an example of mathematical structuralism or not, and so forth. Lot of fancy philosophizing :-)

And they point out some of the drawbacks with structuralism. If the natural numbers are not any particular collection of objects, but rather are instances of some "structure," then what exactly is a structure?

The main point is that when we say that 0 is the empty set and 1 is the set containing 0, what we really mean is that these sets represent the natural numbers within set theory. What we don't mean is that these sets actually "are" the natural numbers. Leaving unanswered the question of what the natural numbers really are

I found the comments about Cohen's Filter in the article I linked fascinating. Like most math people I knew of his breakthrough results, but was unfamiliar with the actual math. I'd be interested to hear opinions from the set theorists on the forum about this. — jgill

I don't think there are any set theorists here. You're the only mathematician in the house. The rest of us, speaking for myself, are groupies and hangers-on at best. That said, can you say what "this" refers to? Cohen's invention of forcing in general? Or the particular recent result that's floating around the Internet about Martin's Maximum implying (*) or some such? The latter is some serious set-theoretic inside baseball.

Could someone rightfully say that 0, 1, and points are not in any sense sets? Or is there more too that? — Gregory

Sure. Euclid didn't have set theory but he talked about points. As far as the modern definition of numbers, there's Russell's type theory and its modern variants, there are category-theoret definitions, and so forth. I don't know much about any of these alternatives.

Benacerraf described it like this:

To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4,5, and so forth. And to be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by.... Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role -not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression.

That is, the number 3 is not an object at all. Rather, it's a thing defined by its relation to other numbers. In his famous essay he kicked off the field of mathematical structuralism. @TonesInDeepFreeze already gave this link, I'll repeat it for reference.

https://plato.stanford.edu/entries/structuralism-mathematics/

Benacerraf's essay can be downloaded here. The quote above is found on page 70,

https://documents.pub/document/benacerraf-what-numbers-could-not-be.html

You have to click Download then it makes you wait 60 seconds. Other online links to the article either make you read it online or else don't let you read it at all. When I'm in charge, academic paywalls will be abolished. Taxpayers already paid for this research. I looked it up. Benacerraf worked at Princeton and Princeton takes Federal money.

In category theory there's a thing called a natural numbers object which is intended to capture the structural essence of natural numbers. I don't know much about this and the Wiki article isn't particularly enlightening.

Here's an article about the natural numbers type in modern type theory. It's also not very enlightening unless one is a specialist.

So the bottom line is that structuralists don't think that natural numbers "are" sets; or even that natural numbers are any particular thing at all. A natural number is whatever relates to other things the way natural numbers do.

ps -- Here's the Wiki article on mathematical structuralism.

https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)

I think you meant 'transitive set well ordered by ∈'. — TonesInDeepFreeze

Thanks, I made the correction.

OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser.. — Metaphysician Undercover

Ok good.

I assume that an ordinal is a type of set then. — Metaphysician Undercover

Yes. But that should be no surprise. In set theory everything is a set. There are no urelements in standard set theory. In math every single thing is a set. Numbers, groups, topological spaces, cardinals, ordinals, are all sets. Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. There is nothing but sets. Of course one need not found math on set theory, but in standard math, that's how it's done. Everything is a set.

It consists of identifiable elements, or parts, some ordinals being subsets of others. — Metaphysician Undercover

Yes. And those elements are sets. and those sets' elements are sets, all the way down to the empty set. Everything is a set. That's why they say math is based on set theory. Of course that's only historically contingent. Are numbers "really" sets? That's the question raised (and answered in the negative) by Paul Benacerraf in his famous essay, What Numbers Could Not Be.

My question now is, why would people refer to it as a "number"? — Metaphysician Undercover

There's no general definition of number. Negative numbers didn't use to be regarded as numbers, nor did zero, irrational numbers, complex numbers. Quaternions are numbers these days, but William Rowan Hamilton got famous for discovering/inventing them in 1843. When Cantor introduced his cardinals and ordinals he got a lot of pushback from the mathematical community of the day, but in the end his point of view won out, and the transfinite ordinals and cardinals are numbers. What is a "number" is a matter of historical contingency.

Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. — Metaphysician Undercover

Ok. And lets be perfectly explicit. In the formalism, $0 = \emptyset, 1 = \{0\}, 2 = \{0, 1\}, 3 = \{0, 1, 2\}$ and $4 = \{0,1,2,3,\}$. Of course this is only a formalism. As Benacerraf points out, the number 4 can't really "be" this set. Rather, it's just a particular representation. The number 4 is the abstract thingie pointed to by the representation. But we've had this conversation before.

By what principle is this group of elements united to be held as an object, a number? — Metaphysician Undercover

Groups of elements are united to be held as sets by the axioms of set theory. If X is a set and Y is a set then their union and intersection are sets, and so forth. You can find the axioms here.

Now what is an "object," I don't know, because object is a term of art in computer programming but not in math. And what's a number is, as I've pointed out, a matter of historical contingency. There are no principles other than Planck's great observation that scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks grow up taking the new ideas for granted.

Do you know what I mean? — Metaphysician Undercover

Yes. You want to know what entitles $\omega^\omega$ to number-hood. Well it's the same thing that entitled $\sqrt 2, i$, and -47 to numberhood. Historical contingency. Someone said "Hey this weird thing is useful, I'll call it a number," and everyone else said, "You're crazy," and a generation or two later everyone called it a number. Simple as that. Human opinion over time, nothing deeper than that.

A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. — Metaphysician Undercover

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. The concept of "set" itself has no definition, as I've pointed out to you in the past. A particular set might have a specific definition; but even that sometimes fails, as in the nonconstructive sets given by various set-theoretic axioms. A set exists when the axioms say it does. To take a non-mathematical example, the set consisting of the number 5, the tuna sandwich I had for lunch, and the Mormon Tabernacle Choir may be taken together into a set consisting of three elements. Of course in math you can't have examples like that; the elements of sets have to be other sets. Unless you are working in a set theory that has urelements, which is a bit of a niche area.

In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set? — Metaphysician Undercover

There is no set of ordinals, this is the famous Burali-Forti paradox.

What makes a particular set an ordinal is that it satisfies the textbook definition of an ordinal, namely a transitive set well-ordered by $\in$. That technical definition needs to be unpacked, but that's the definition. If a set satisfies that definition, it's an ordinal.

The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number". — Metaphysician Undercover

I've been pointing out to you the different meaning of cardinal equivalence and cardinal number for several posts now. I'm not sure why you claim I am not "acknowledging" that difference. I have been expending quite a few keystrokes to explain that distinction to you.

Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting. — Metaphysician Undercover

I get the point you're making, it's an interesting philosophical point. If I define an odd number as a number that leaves a remainder of 1 when integer-divided by 2; and I then prove that 47 is an odd number; was 47 an odd number before I made the definition? It's a good question in mathematical philosophy. Not one we'll solve here today.

Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover

Well not exactly. An ordinal is a type of set, yes. But a cardinal is not a "type" of ordinal at all. Rather, among all the ordinals cardinally equivalent to a given set, we take the least of them and designate that as the set's cardinal. So the cardinal-ness of an ordinal is not a property of an ordinal; rather, it's a name we give to an ordinal that has a particular property relative to a lot of other ordinals. Subtle point but important. It's a little like the captain of a football team. The captain is not a "type" of player; rather, the captain is a player that we have designated as the captain. The ordinal definition of cardinal is like that.

Logical priority is given to "set". — Metaphysician Undercover

In the sense that in the set-theoretic formalization of math, sets are fundamental. Ok. If that's what you're saying.

So do you agree that a cardinal number is not an object, but a collection of objects, as a set? — Metaphysician Undercover

I don't know what an object is (except in the context of everyday English, or computer programming; but not in math); so you'll have to tell me.

But a cardinal number is a set, yes. Everything is a set in set theory. Everything is a set.

Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers? — Metaphysician Undercover

I have no idea what you mean by object. I only know about sets. The defining principles of what can be called sets are the axioms of set theory. What's called a number is a matter of human agreement, often hard-won over generations and always subject to revision.

Even the same object in different contexts is or isn't a number. A classic example is the number $i$, the imaginary unit. We call that a number. But we can model the complex numbers as a particular set of 2x2 matrices, and then we call them matrices and not complex numbers. The question of what's a number is a matter of human convention. There is no general definition of number.

But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. — Metaphysician Undercover

Well of course that's an interesting philosophical question, which we will not solve today. Was 47 an odd number before I defined what an odd number is? You want to say that somehow the cardinal numbers existed before we defined them. Fine, you're a Platonist today. Sometimes I am too. Other times, not so much. What of it? I agree it's a good question. Whether the cardinal numbers were "out there" waiting for von Neumann to come along and give them their definition; or whether he made them up out of his productive mind.

After all, in other posts you have cast personal doubt on the very existence of mathematical sets; and now you want to claim that cardinal numbers were already out there waiting for von Neumann to come along. You see you're at best a part-time Platonist yourself.

You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count. — Metaphysician Undercover

If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing.

And then I'll ask, well if the number 5 existed before the universe did, where did it exist? What else might live there? The Baby Jesus? The Flying Spaghetti Monster? Captain Ahab? Platonism is hard to defend once you start thinking about it.

I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it?

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself.
— fishfry

But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made." — Metaphysician Undercover

Bit disingenuous there. The relation "has the same parents" is a binary relation, it inputs two people and outputs True or False. But they don't have to be distinct people. I have the same parents as myself.

Yes this exemplifies the ontological problem I referred to. — Metaphysician Undercover

Which I fully acknowledge, and note that we are not going to solve it here. Were the transfinite cardinals out there waiting to be discovered by Cantor and then formally defined by von Neumann?

But I must note that I find it very strange to see you suddenly advocating for mathematical Platonism, after denying the existence of mathematical sets.

Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"? — Metaphysician Undercover

Heck of I know. Did the number 5 exist before the Big Bang? Was it out there waiting for humanoids with five fingers to come along? Maybe you can answer me that first, before you ask me about the transfinite cardinals.

I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement. — Metaphysician Undercover

I will agree that the fiveness of the fingers on my hand is not as arbitrary as my assignment of categories to people such as which room I put them in at a party, or who I designate as the captain of the team.

But @Meta, really, you are a mathematical Platonist? I had no idea.

If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me. — Metaphysician Undercover

I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again.

Yes you are correct, it's cleaner to not use proof by contradiction.
— fishfry

Is that a thing? Ok. — bongo fury

It's perhaps a little cleaner in terms of exposition. Not a big deal either way.

OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity? — Metaphysician Undercover

I answered this in my most recent post to you. Given two ordinals, it's always the case that one is an element of the other or vice versa. So for ordinals $\alpha$ and $\beta$, we define $\alpha \lt \beta$ if it happens to be the case that $\alpha \in \beta$. This is perfectly well-defined and unambiguous, especially in the case of the von Neumann ordinals which are constructed exactly so that this works out. $\emptyset \in \{\emptyset \} \in \{{\emptyset, \{\emptyset\}\}} \in \dots$, and that's exactly how ordinal "less than" is defined. It's also the case that the von Neumann ordinals are defined in such a way that $\in$ is transitive; if $\alpha \in \beta$ and $\beta \in \gamma$ then $\alpha \in \gamma$. This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work.

Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity? — Metaphysician Undercover

Via the $\in$, which is a primitive in set theory and is always true between any two ordinals in one direction or the other, by construction of the ordinals.

Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work. — Metaphysician Undercover

The Wikipedia articlea on ordinals, cardinals, Cantor, etc. are perfectly satisfactory in this regard.

OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying? — Metaphysician Undercover

Rather than try to save that example, I'll just repeat the hand/glove example. By putting on a glove, I can determine whether my hand-fingers are in bijective correspondence with the glove-fingers, without knowing the actual cardinal number.

c
Here's where the problem is. You already said that there is a cardinality which inheres within ordinals. [/quote]

I went to great pains to note last time that your use of the word "cardinality" is ambiguous and causing you to be confused. You should either say cardinal equivalence or cardinal number, to clearly disambiguate these two distinct but related notions. Every ordinal number is inherently cardinally equivalent to many sets. If nothing else, every ordinal is cardinally equivalent to itself, so the point is made.

But we still don't know which cardinal number that is.

And again, when you say "cardinality," you obfuscate the distinction between these two concepts.

This means that cardinality is a property of all ordinals, [/qmote]

No no no no no. No. Every ordinal is cardinally equivalent — Metaphysician Undercover

to many other sets, including itself. But when we clarify this terminology, your sophistic point evaporates.

it is an essential, and therefore defining feature of ordinals. — Metaphysician Undercover

No, as I'm pointing out to you. It's true that every ordinal is cardinally equivalent to itself, but that tells us nothing. You're trying to make a point based on obfuscating the distinction between cardinal numbers, on the one hand, and cardinal equivalence, on the other.

So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals, — Metaphysician Undercover

No no no no no. I hope I've explained this.

and we also have a sense of "cardinal" number which is specific to a particular type of ordinal. — Metaphysician Undercover

Other way 'round. A cardinal number is defined as a particular ordinal, namely the least ordinal (in the sense of set membership) cardinally equivalent to a given set.

I hope you can see that by carefully using the phrases, "cardinal equivalence" and "cardinal number" properly, all confusion goes away. You are deliberately introducing confusion by using the word "cardinality" ambiguously.

Don't you see how this is becoming nonsensical? — Metaphysician Undercover

No, it's very carefully thought out by 150 years worth of mathematicians including von Neumann, widely agreed to be "the smartest man in the world" by his contemporaries. It's your insistence that everyone else is wrong about things that you aren't willing to put in the work to understand that's nonsensical.

What you are saying is that it has a cardinality, — Metaphysician Undercover

No. I am repeatedly telling you to stop using that word, because you are using it to confuse yourself. It's true that every ordinal is cardinally equivalent to various other sets including itself; and it's true that a cardinal number is defined as a particular ordinal.

Your entire argument is based on obfuscating the word cardinality. You should stop, because I can only explain this to you so many times without losing patience. I already explained this to you repeatedly in my previous post.

because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number. — Metaphysician Undercover

Right. I can live with that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is.

In essence, you are saying that it both has a cardinality, because it is cardinally equivalent, — Metaphysician Undercover

No. I'm telling you to stop using the word cardinality until you understand what's being talked about.

and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number. — Metaphysician Undercover

This is you just continuing to confuse yourself over the word cardinality. If you'll just carefully say cardinal equivalence when you mean that, and cardinal number when you mean that, we might make progress.

Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it? — Metaphysician Undercover

Because cardinal numbers are a defined term[/url]. Given a set, we have to build a sophisticated technical apparatus in order to define what we mean by its cardinal number.

But I could take a step back from all this. My remark about what's logically prior to what is true, but it's not that important in the scheme of things. It's more important for you to make an effort to understand what ordinal numbers are, because they're important. So if all you care about it to be right about the logically prior business, that's the wrong thing to care about. It's not an important matter.

Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner? — Metaphysician Undercover

Given two ordinal numbers, it's always the case that one is an element of the other, as sets. We define $\lt$ as $\in$. If you prefer you can always think of the $\in$ whenever I say that one ordinal is "less than" another, or that some ordinal is the "least" with such and so property.

Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, — Metaphysician Undercover

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. So it is not true that "a set has a cardinal equivalence" in isolation. That makes no sense. I can say, "My brother and I have the same parents," but it makes no sense to say, "I have the same parent." As what?? Having the same parent is a relation between two things. It doesn't stand alone. Likewise two sets may be cardinally equivalent to each other or not. But a set doesn't have cardinal equivalence by itself, that makes no sense.

it also necessarily has a cardinality, — Metaphysician Undercover

Please stop using that word till you have a better grasp on the material. Your constant misuse of it is only causing you confusion.

and a corresponding mathematical object which you call a cardinal number. — Metaphysician Undercover

Yes, it has that, after we've built up a whole bunch of theory to define what that is.

Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number? — Metaphysician Undercover

Because after Cantor defined cardinal equivalence, the question came up among philosophers, "What actually is a cardinal?" At first they did the obvious thing, they said a cardinal was the entire class of all the sets that are cardinally equivalent to a given set. That is a perfectly satisfactory definition, but it suffers from the flaw that such a class is not a set. Von Neumann figured out how to define cardinal numbers as particular sets, so that they could be manipulated using the rules of set theory.

It's just a matter of wanting to define what a cardinal number is, in formal terms. So that when we have two sets that are cardinally equivalent to each other, we have a collection of canonical sets such that every set is cardinally equivalent to exactly one of them.

I see where you're going with this. Given a set, it has a cardinal number, which -- after we know what this means -- is its "cardinality." You want to claim that the set's cardinality is an inherent property. But no, actually it's a defined attribute. First we define a class of objects called the cardinal numbers; then every set is cardinally equivalent to exactly one of them. But before we defined what cardinal numbers were, we couldn't say that a set has a cardinal number. I suppose this is a subtle point, one I'll have to think about.

Here's an example. Whenever I have a party I like to put everyone in separate rooms according to their approximate height. I have ten rooms and I arrange the people so that there's more or less an equal number of people in each room. So at the party, each person is a "room 1 person" or a "room 2 person" and so forth.

But when you got up that morning, before you came to my party, you weren't a room 3 person or whatever. The assignment is made after you show up, according to a scheme I made up. Your room-ness is not an inherent part of you.

Likewise, given a set we can assign it a cardinal number. But it's far from clear that this is an inherent property of a set. Rather, we set up the scheme of defining cardinal numbers so that given a set, we can figure out which cardinal it's assigned to. It's an after-the-fact defined assignment, not an inherent quality.

Another example, a bunch of people show up for a work detail. I assign some to dig ditches, some to cut down trees, some to supervise, some to do this, some to do that.

Before I made the work assignments, the jobs were not inherent properties of the people. Rather, I assigned those jobs after the fact. Just as I can take a set and assign it a cardinal number. But of course in the case of a set's cardinal number, that's a more subtle question. Did the set "inherently" have a cardinal number before I assigned it?

Good point, if that is your point. I'll give this some thought.

Or can ripples or motion itself exist in this ocean? — TiredThinker

If you drop a pebble in the ocean it will ripple forever. And if you drop lots of pebbles there will be lots of ripples, all the time.

With my original question I didn't think too hard on the point that if one did take a drop of water from an infinite ocean they would have no place to take it. And if I created infinite land next to an infinite ocean that might create even more questions. — TiredThinker

In general, a set can be infinite yet not include "everything." For example there are infinitely many even numbers, but they don't include the odd numbers.

In the case of an infinite ocean we have to work a little harder to get a good visualization. Maybe the world is like a 3D chessboard, with an infinite ocean on one level and an infinite plane of land on the next. So we can be sitting on land and reach down to take a cup of water from the infinite ocean below. Now we have a cup of water; but since the ocean is liquid, it immediately fills up the space where we removed the water, and there's still an infinite ocean.

What do you think? Visualization-wise, I mean?

The essential idea though is that you can always take a finite amount from an infinite set, and the set is still infinite. But it doesn't necessarily have to be all of what it was before. I believe @TheMadFool gave this example earlier, where we can start with the infinite set 1, 2, 3, 4, ..., then remove 1 to leave 2, 3, 4, ... What's left is still infinite, yet it's missing 1. That can happen too. Infinity is funny that way.

ps Here's another idea. The world is a flat, infinite plane. Like "flat earth" theory except instead of a great wall of ice around the edge, it just goes on forever. The entire world is a vast ocean, but there are infinitely many finite-sized islands spread throughout. So there's an island here and another one there, infinitely many in all, but they're all separated by water. So if you're on land you can always dip your cup in the infinite ocean.

We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze

Yes you are correct, it's cleaner to not use proof by contradiction. Thanks for the clarification.

Wow that's simply put. Thanks — Gregory

You are welcome! So glad that worked for you. This argument is much simpler and more natural than the diagonal argument, it should be better known.

With the rise of "Far Right Extremism" and the Right so emboldened as to storm the Capitol building, where is the Far Left's Uprising? There are forms of Left Wing radicalism apparent in the US, but none so apparently emboldened as the Right's. — Lil

You missed the Antifa and BLM "mostly peaceful" riots last summer in which at least 23 people were shot dead, 700 police officers injured, 150+ federal buildings damaged, hundreds of small businesses destroyed, and billions of dollars in property damage was done?

No, I don’t think so and for the same reason I stated. I don’t know of any solution, but there has to be a better alternative than aggrandizing the state.

It was government posturing and regulations that led to censorship on social media in the first place. — NOS4A2

As a fellow libertarian I salute your absolutism and ideological purity. I myself take a more nuanced approach. Lunch counters have to serve everyone and the health department has the right to make sure the local restaurants are keeping out the rats. And social media companies serving as de facto town squares can't engage in what's legally called viewpoint discrimination. And as has been noted, even a private company can't violate the Constitution when it is effectively acting as a government agent, as Jen Psaki just admitted Facebook is doing.

I've asked people many times and they bring up the diagonal thing, although this just shows there are infinity more uncountable than countable and yes, however there are infinity many natural than odd. But you can biject with one and not the other? I'm not a jerk, just want some way I can understand what they are saying. It seems to me infinity is always just infinity at the end — Gregory

Have you seen the simple and beautiful proof of Cantor's theorem? It shows that there is no possible surjection from a set to its powerset.

Here's the proof. Let $X$ be a set, $\mathscr P(X)$ its powerset, the set of all subsets of $X$. Let $f: X \to \mathscr P(X)$ be a function, and by way of starting a proof by contradiction, suppose $f$ is a surjection.

Since $f$ is a function that inputs an element of $X$ and outputs some subset of $X$, for any given element $x \in X$ it may or may not be the case that $x \in f(x)$. Let $S$ be the subset of $X$ defined by $S = \{x \in X : x \notin f(x)\}$.

Now $S$ is a subset of $X$; and since by assumption $f$ is a surjection, there must be some element $s \in X$ such that $f(s) = S$.

Now we ask the question: Is $s \in S$? Well if it is, by definition it isn't; and if it isn't, by definition it is. Therefore the assumption that there is such an $s$; that is, that $f$ is a surjection; leads to a contradiction.

Therefore there is no surjection from any set to its powerset.


To take an example, consider the set {a,b,c}. Its powerset is {ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}. It's perfectly clear that there's no surjection from a 3-element set to an 8-element set; and this principle holds even in the infinite case.

Cantor's theorem immediately gives us an endless hierarchy of infinities $\mathbb N, \mathscr P(\mathbb N), \mathscr P( \mathscr P(\mathbb N)), \dots$ In terms of the simplicity of the argument versus the profundity of the result, I don't think there's anything comparable in all of mathematics.

I, for one, don’t want to live in such a society. I believe giving the state such power has the corresponding effect of diminishing social power. — NOS4A2

Do you think the government should have the power to inspect restaurants and shut them down if it finds them operating contrary to the public interest? Or do you favor a "buyer beware" policy where if enough customers drop dead, everyone else will eat somewhere else?

This is not intended to be a difficult question. It's meant to make the point that even a libertarian believes in the board of health. Or eats at home.

https://caitlinjohnstone.com/2021/07/16/biden-administration-completely-kills-the-its-a-private-company-so-its-not-censorship-argument/ — StreetlightX

Caitlyn got that talking point from me :-)

I do disagree. — NOS4A2

I don't think anyone wants to live in a society where the government can't restrict the behavior of private companies. How about this example. Do you think the local board of health should be allowed to inspect privately-owned restaurants, and shut them down if they are operating in an unsafe or unclean manner?

The dictum “they are private companies” holds true. When the government forces a company such as Facebook to operate in an approved manner, it violates their free speech. — NOS4A2

Do you disagree with calling privately owned lunch counters public accommodations in order to force them to serve black customers? They're private companies too, and before the Civil Rights Act of 1964, it was perfectly legal for them to discriminate on the basis of race.

SO do you folk agree that opposing the vaccine is a bad thing? — Banno

The subject is free speech. One can support X and still support the right of people to express opposition to X. When you support X and also support suppressing the free speech rights of people who disagree with you, that crosses a line to a very bad place that nobody wants to go.