Turley is definitely not a "liberal". The article you linked doesn't actually analyze the decision, it just asserts that it is correct, and then procedes to chastize liberals whoCdisagree with the decision. — Relativist

He's written much more detailed legal analyses that I probably had in mind, if that particular article may have been too general.

He's not a liberal because he is a liberal who sees the recent misbehavior of liberals, and talks about it. So he gets called a conservative, and he appears on FOX and in the New York Post because Rachel Maddow and the New York Times won't speak to him anymore.

Liberal legal scholar Alan Dershowitz (controversial for other reasons) talks about this. He is a lifelong liberal Democrat. He defended Trump in the US Senate, and he has complained that now his Martha's Vineyard friends and neighbors won't talk to him. He said he never got this kind of response when he defended murderers.

Glenn Greenwald is another, a lifelong liberal who wrote for Salon and called out Obama's foreign policy as being an extension of Bush's. At the time I had the same impression, and reading Greenwald kept me sane. I wasn't the only one who saw what I saw. Today of course Greenwald gets the "not a liberal" appellation from liberals who don't like it when anyone has a heterodox opinion.

Comedian Jimmy Dore is another one, a Bernie supporter tarred as a right winger by the same illiberal liberals. Many other alt-media figures. Matt Taibbi, smeared as a "so-called journalist" by Dem Rep. Stacey Plaskett in the House social media censorship hearings. So many more. Any liberal who doesn't toe the line gets smeared as a right winger.

I do apologize if Turley's article wasn't detailed enough. I just skimmed it and relied on other much more detailed arguments about the case that Turley has made, and also Dershowitz. Elie Honig on CNN had the same take, that it's a bad case. Many other liberals have made that point. Of course once they criticize Bragg's case they're "not really liberals," as you illustrate.

You stuck with me for an incredibly long time, and while I wasn't specifically looking for a sounding board, it turns out that's exactly what I needed. However, we've now reached a point where this discussion requires more than just a sounding board; it needs someone who can truly digest and engage with what I'm saying. Given this, it makes sense for you to disconnect now. This isn't due to a lack of topics to discuss, but rather because you're not interested. As such, I don't foresee us picking up this conversation again in the future, but who knows. Thank you so much for staying with me up to this point. I've gained a lot from our discussions, and I wish you all the best! — keystone

You're very welcome. You know you had a profound effect on my life. I had been away from this board for quite some time, a couple of years or more I think. I did read it on occasion; and when you posted your original question in this thread, I jumped in because I happened to know that the answer to your puzzle is that there is no uniform probability on a countably infinite set. Your top post lured me back from my vacation. For what that's worth. The jury's still out or maybe they're measuring the rope.

I enjoyed our chat. Thank you.

There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this. — ssu

I agree that's suitable for this forum. Just not for "Intro to Math," which I interpreted as "Last math class the liberal arts majors will take," or something like the Discrete Math class they teach these days to math and computer science majors.

Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.) — ssu

That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.

I don't think they teach basic arithmetic anymore. It's a problem in fact.

It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell. — ssu

There's always Gödel's Proof by Nagel and Newman. And Gödel, Escher, and Bach: An Eternal Golden Braid by Hofstadter. Actually I only leafed through it once but everyone raves about it. I'm not up on the literature of pop-mathematical logic. Or real mathematical logic, for that matter.

No, Tones was referring to the principle called "the identity of indiscernibles", which is completely different from the law of identity. The law of identity makes a thing's identity the thing itself, the identity of indiscernibles associates a thing's identity with the thing's properties. These are fundamentally different principles. — Metaphysician Undercover

Yes I understand that. I didn't realize @TonesInDeepFreeze was talking about IofI. Actually I just learned there's an identity of indiscernibles and an indiscernibility of identicals.

No, you simply fell for the sophistry. Tones is very good at it, and apt to convince others, earning the title "head sophist". — Metaphysician Undercover

Ok this is interesting. My quote was, "Sets are subject to the law of identity." So that if X is a set, I can say X = X without appealing to any principle of set theory.

Tones convinced me of that. Now you say that he only sophisted me. How so please? If X is a set, how is X = X not given by the law of identity? You have me curious.

You think I'm a victim of Tones's sophistry. That is an interesting remark.

So, a set is a mathematical structure. — Metaphysician Undercover

Set theory is a mathematical structure. The analogy is:

Set theory is to group theory as a particular set is to a particular group.

But a set is a mathematical structure too, since the elements of sets are other sets.

How do you make this consistent with the head sophist's claim that the members of The Beatles is a set, and the particles which make up a rock is a set? The sophist says "the set whose members are all and only the bandmates in the Beatles has 24 orderings". Notice that this is not stated as possible orderings, it is stated as the "orderings" — Metaphysician Undercover

I take no responsibility for anyone else's posts. I barely take responsibility for my own. You've already told me you don't like real world examples about playgrounds so I don't use those anymore with you.

Remember your schoolkid example? — Metaphysician Undercover

Yes. I agreed with you that this was only a casual analogy, and once you told me that you don't like it, I stopped using it.

You recognized that the objects which bear that name have what you called SOME order, and this is an expression of the condition which they are actually in, at any point in time. I would call this their "actual order". — Metaphysician Undercover

This is true about kids in playgrounds, NOT mathematical sets. You have informed me that you don't like real-world analogies so I no longer use them. Mathematical sets have no inherent order.

No, Tones was referring to the principle called "the identity of indiscernibles", which is completely different from the law of identity. The law of identity makes a thing's identity the thing itself, the identity of indiscernibles associates a thing's identity with the thing's properties. These are fundamentally different principles. — Metaphysician Undercover

Yes I understand that. I didn't realize @TonesInDeepFreeze was talking about IofI. Actually I just learned there's an identity of indiscernibles and an indiscernibility of identicals.

No, you simply fell for the sophistry. Tones is very good at it, and apt to convince others, earning the title "head sophist". — Metaphysician Undercover

Ok this is interesting. My quote was, "Sets are subject to the law of identity." So that if X is a set, I can say X = X without appealing to any principle of set theory.

Tones convinced me of that. Now you say that he only sophisted me. How so please? If X is a set, how is X = X not given by the law of identity? You have me curious.

You think I'm a victim of Tones's sophistry. That is an interesting remark.

So, a set is a mathematical structure. — Metaphysician Undercover

Set theory is a mathematical structure. The analogy is:

Set theory is to group theory as a particular set is to a particular group.

But a set is a mathematical structure too, since the elements of sets are other sets.

How do you make this consistent with the head sophist's claim that the members of The Beatles is a set, and the particles which make up a rock is a set? The sophist says "the set whose members are all and only the bandmates in the Beatles has 24 orderings". Notice that this is not stated as possible orderings, it is stated as the "orderings" — Metaphysician Undercover

I take no responsibility for anyone else's posts. I barely take responsibility for my own. You've already told me you don't like real world examples about playgrounds so I don't use those anymore with you.

Remember your schoolkid example? — Metaphysician Undercover

Yes. I agreed with you that this was only a casual analogy, and once you told me that you don't like it, I stopped using it.

You recognized that the objects which bear that name have what you called SOME order, and this is an expression of the condition which they are actually in, at any point in time. I would call this their "actual order". — Metaphysician Undercover

School kids and physical objects in general do. Mathematical sets don't. You explained to me that you don't like physical examples so I no longer use them. Mathematical sets have no inherent order. The purpose of defining things that way is so that we may study the abstract notion of order.

Can you see what the head sophist has done? The sophist has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same, being in each case a different presentation of the same set. But you and I recognize, that in reality there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time. The sophist might talk about 24 orderings, but you and I recognize that if these 24 account for all the possibilities, only one of those possibilities represents the very special "actual order", and, that since these elements are physical objects, there must be an actual order which they are in, at any given time. — Metaphysician Undercover

I can't comment on what anyone else has said. This has nothing to do with the conversation you and I are having.

The law of identity is very important to recognize the actual existence of a thing, and its temporal extension. — Metaphysician Undercover

A temporal extension. You are saying it only applies to things that exist in time? Meaning not sets? I don't think that's right. Any set is identical to itself and also equal to itself by virtue of the law of identity.

Tones did explain that to me, but not via sophistry. He asked me to prove the transitivity of set equality. Once I attempted to do that, I realized that I needed not the axiom of extensionality, but its converse. And that converse is true by way of the law of identity from the underlying predicate logic. This I discovered for myself when Tones pointed me to it.

Through time a thing changes, and the law of noncontradiction stipulates that contradicting properties cannot be attributed to the same thing at the same time. So if a specific group has ordering A at a specified time, that is a property of that group, and it surely cannot have ordering B at the same time. The head sophist claims that the specified group has 24 orderings, all the time (as time is irrelevant in that fantasy land of sophistry). Obviously the head sophist has no respect for the law of noncontradiction, and is just making contradictory statements, in that sophistic fantasy. — Metaphysician Undercover

That may or may not be true about physical objects. You say the kids in height order is not identical to the kids in alphabetical order. I say the set of kids is the same. But I do not argue this point and od not care about it.

I tell you that a set has no inherent order; and that the set of natural numbers in its usual order; and the set of natural numbers in the even-odd order say -- 0, 2, 4, 6, ...; 1, 3, 5, 7, ... is exactly the same set. It is a different ordered set, because in an ordered set, the order is part of the identity of the set. In a plain set, it's not. This is how mathematicians play their abstraction game.

That is what happens when we allow that abstractions such as mathematical structures have an identity. Inevitably the law of noncontradiction and/or the law of excluded middle will be violated. Charles Peirce did some excellent work on this subject. It's a difficult read, and you've already expressed a lack of interest in this subject/object distinction, so you probably don't really care. Anyway, here's a passage which begins to state what Peirce was up to. — Metaphysician Undercover

On the contrary, I've expressed great interest in the ideas of Pearce when members of this forum have mentioned them to me.

The relevance of all this to the principles of excluded middle and contradiction is as follows. Peirce wrote that “anything is general in so far as the principle of excluded middle does not apply to it,” e.g., the proposition “Man is mortal,” and that “anything” is indefinite “in so far as the principle of contradiction does not apply to it,” e.g., the proposition “A man whom I could mention seems to be a little conceited” (5.447-8, 1905). If we take Peirce to have meant LEM and LNC, then it appears that he wanted to deny the principle of bivalence (according to which all propositions are true or else false) with regard to universally quantified propositions, and that he meant to claim that existentially quantified propositions are both true and false. But why think that “Man is mortal,” which seems to be straightforwardly true, is neither true nor false? And why think that one and the same proposition, “A man whom I could mention seems to be a little conceited,” is both true and false? Once we see what Peirce meant by “principles of excluded middle and contradiction,” we see that this is not what he was claiming.
— Digital companion to C. S. Peirce — Metaphysician Undercover

Yes, well, discussions of denying LEM don't interest me much. I'll agree with that. But I've come by it honestly. I've made a run at constructivism and intuitionism more than once. I've read Andrej Brauer's "Five Stages of Accepting Constructive Mathematics." It doesn't speak to me. The paragraph you quoted is a little above my philosophical pay grade. Perhaps you can explain its relevance to the topic at hand.

http://www.commens.org/encyclopedia/article/lane-robert-principles-excluded-middle-and-contradiction — Metaphysician Undercover

I'll take a look as the spirit moves me, but I don't think this is a particularly productive line of conversation for me. I don't know what you are trying to tell me.

This is blatantly untrue, and as demonstrated above, if we assign "identity" to a set, the law of non-contradiction will be violated. — Metaphysician Undercover

I don't see why. If X is a set, then X = X by identity.

Now if you are trying to say that the order properties might differ or whatever, I say you are just being your usual anti-abstraction self. I don't understand your aversion to mathematical abstraction. But it doesn't effect my mathematical ontology in the least.

Or maybe you're saying something else. If so, please explain.

The law of identity enables us to understand an object as changing with the passing of time, while still maintaining its identity as the thing which it is. — Metaphysician Undercover

There is no time in set theory. Mathematics is outside of time, or talks about things that are outside of time.

Sets have distinct formulations existing all the time, which would cause a violation of the law of noncontradiction if we allow that a set is subject to the law of identity. Therefore we must conclude that sets are not subject to the law of identity. The type of thing which the law of identity applies to is physical objects. And there is obviously a big difference between physical objects and sets, despite what head sophist claims. — Metaphysician Undercover

You are just wrong about this. But give me a more specific example, if you would, so that I may understand you better.

You also have no problem with contradiction, it seems. — Metaphysician Undercover

I can always tell when you can't defend your point. The insults come out. You can do better, can't you?

This tells me nothing until you explain precisely what ∈ means. [/math]

$\in$ is an undefined primitive of set theory. Its behavior is defined by the axioms.
— Metaphysician Undercover

To me, you are simply saying that x is an element of y if x is an element of y. What I am asking is what does it mean "to be an element". — Metaphysician Undercover

It doesn't "mean" anything. It's an undefined primitive in set theory, as point, line, and plane are undefined primitives of Euclidean geometry. Its behavior and usage are defined by the axioms.

If we go with this definition, you ought to se very clearly that sets, as categories, abstract universals, do not have an identity according to the law of identity. A category is not a thing with an identity. — Metaphysician Undercover

I have no problem with that. If you want to say that set theory, as the universal, is not subject to identity, that's fine. I can' say that "set theory equals set theory." I'm perfectly fine with that. Nobody ever says it.

But given particular instances of set theory; that is, sets; we can ask if they are equal to each other or not.

So I promise not to say that the universe of sets is equal to the universe of sets. Though the category theorists will probably disagree with you.

Obviously this does not work. As you said already, elements are often sets. Therefore you cannot characterize the set as an abstract universal, and the element as an abstract particular, because they're both both, and you have no real distinction between universal and particular. — Metaphysician Undercover

You are distorting what I said. ANY particular set is a particular instance of the concept of set, as any particular apple is an instance of the concept (or category) of apple. That causes no problem.

You are willfully obfuscating this point. It's a very clear point. The set of integers is a particular. Sets in general, or the concept of sets, are generalities or a category or a universal.

There's no point in trying to justify the head sophist's denial of reality. — Metaphysician Undercover

It's pointless to keep referring to conversations you've had that I haven't seen, with people who aren't me.

If "Cinderella" refers to a particular, an instance of the category "fairy take characters", then that is a physical object. — Metaphysician Undercover

Clearly Cinderella is not a physical object. That's exactly why I used a fictional character as an example.

If "Cinderella" refers to a further abstract category, like in the case of "red is an instance of colour", then it does not refer to a particular. The head sophist seems to have convinced you that you can ignore the difference between a physical object and an abstraction, but you and I both know that would be a mistake. — Metaphysician Undercover

Knock it off, will you please? Take your complaints WITH the other party, TO the other party.

I have long ago agreed that physical objects and mathematical objects are not the same.

Cinderella refers to the individual fictional character of Cinderella, as Captain Ahab refers to the fictional character of Captain Ahab.

No. I'm complaining that things I have never said - and do not know enough mathematics to articulate - have been attributed to me.

All I know is that a link to a message that I did post, on the Infinite Staircase thread, has been added to text and quoted by fishfry in this post ↪fishfry on the Fall of Man thread. Judging by this post ↪fishfry and my exchanges with them on the Fall of Man thread, they did not do this. I don't know who did it and have no way of finding out. The suggestion that it came from keystone came from ↪jgill.

I hope that's clear. I would like the false attribution to be corrected or removed. — Ludwig V

Did I screw up the quoting? Are we having an interesting convo about infinity and time or did I hallucinate that?

I do confess that on this site I process my mentions and don't always know what thread I'm in. It all kinds of blends together. Just for my curiosity, what did I say you said? Like I said earlier, I probably just copy/pasted the wrong quote tag.

It's very helpful, so that's fine. I get my revenge in this post. — Ludwig V

Glad to know. Revenge? What do you mean? By writing a long post? Well I write long posts but prefer when others write shorter ones. I haven't solved this dilemma yet.

The system is not helping me here, because it invites me to link to specific comments, but I'll do my best to make clear what I'm responding to. — Ludwig V

Not sure what you mean. I generally quote the whole post then stick in quote tags around the specific chunks of text I want to respond do.

Perhaps that's why philosophers keep tripping up on them. It is well known that they don't notice what's on the floor - too busy worrying about all the infinite staircases and the fall of man. — Ludwig V

I don't know many philosopher jokes.

That was not a very well thought out remark. I would certainly have hated them in the long-ago days when the Pythagoreans kept the facts secret so that they could sort it out before everyone's faith in mathematics was blown apart. But now that mathematicians have slapped a label on these numbers and proved that they cannot be completed, I'm perfectly happy with them. — Ludwig V

Sorry maybe I was off track about the rationals.

Yes. Austin invented them, Grice took them up, Searle was the most prominent exponent for a long time, although he has now moved on to other things now. — Ludwig V

Yes he got in trouble for harassing his female doctoral students.

It's a thing in philosophy For me, it's a useful tactical approach, but a complete rabbit-hole as a topic. — Ludwig V

Ok. Why did you bring it up relative to math? Oh I remember. "Let x = 3" brings a variable x into existence, with the value 3. So statements in math are speech acts, in the sense that they bring other mathematical objects into existence. I can see that.

Something like that. The initial point was to establish that there are perfectly meaningful uses of language that are not propositions (i.e. capable of being true or false), in the context of Logical Positivism. I doubt that you would welcome a lot of detail, but that idea (especially the case of the knight in chess) will be at the bottom of some of the later stuff. — Ludwig V

Ok. Not entirely sure where you're going.

It was very helpful to me. I have doubts about the terminology "potential" vs "completed", but the idea is fine. I particularly liked "don't really find a use in math". — Ludwig V

Ok I was only trying to be philosophical. Aristotle (I think) made the distinction. It doesn't come up in math, nobody ever uses the terminology. But the way I understand it is that Peano arithmetic is potential and the axiom of infinity gives you a completed infinity.

Too much or not. It helped me. Someone else started talking about bounds and I couldn't understand it at all. I may not understand perfectly, but I think I understand enough. — Ludwig V

Ok, bounds. They're just the shoulders of the road. Thing's you can't go past. Guardrails.

I know that. It's not a problem. If I said anything to suggest otherwise, I made a mistake. Sorry.
— Ludwig V

I understand that distinction. — Ludwig V

That was about limits versus "termination state." I should emphasize that limits are perfectly standard mathematical terminology. But "termination state" is my own locution for purposes of talking about supertasks. The termination state is like a limit in the sense that we can conceptually "stick it at the end" of an infinite sequence; it just doesn't have to satisfy the definition of the limit of a sequence. Like 1/2,/ 3/4, 7/8, ...; 42

The semicolon notation is my own too. I don't think mathematicians talk about supertasks. They're more of a computer science and philosophy thing.

Many of my notions are naive or mistaken. But this separation is my default position. I'm not making an objection, but am trying to point out what may be a puzzle, which you may be able to resolve. On the other hand, this may not be a mathematical problem at all. — Ludwig V

I am not aware of what problem or puzzle you are expressing.

There are other ways of putting the point. What about "Mathematics is always already true"? Or mathematics is outside time? Or time is inapplicable to mathematics? — Ludwig V

The subject matter of mathematics does not speak about time. That's different than saying "math is outside of time," although it's kind of related. Physics talks about time, and physicists use math to model time, but that is a very different thing.

It's the difference between a loop in math versus programming.

In math when we say that 1/2 + 1/4 + 1/8 + ... = 1, we mean "right now," though even that is a reference to timeliness. The equality "just is."

But in a programming language when we write a loop that keeps adding each term to a running total, that notation stands for a physical process that takes place in a computing device and requires time and energy to execute, and produces heat. A programming loop is a notation for a physical process.

But your example of making a rule in chess. Note that as soon as the rules are made, we can starting defining possibilities in chess, or calculating the number of possible games and so forth. It's as if a whole structure springs into being as we utter the words. So a timeless structure is created by our action, which takes place in time. Isn't that at least somewhat like a definition in mathematics? And the definition is an action that takes place in space and time. — Ludwig V

Ok. So when I write down the rules of set theory, I instantiate or create all the complex world of sets as studied by set theorists. And you speculate that this might be an event that takes place in time.

There is another point of view. The structures of the sets were there. Mathematicians discovered the structures. So the discovery of set theory is historically contingent and takes place in time, around 1874 or so with Cantor's first paper on set theory. But the sets themselves, the structures of set theory, are eternal!

In other words this is the old "invented or discovered" question of mathematical philosophy.

Now chess, I think we can agree, was invented and not discovered. But math is somehow different. Math is somehow wired into the logic centers of our minds, and perhaps the universe.

More difficult are various commonplace ways of talking about mathematics. — Ludwig V

Are you referring to what I just talked about?

At first sight, these seem to presuppose time (and even, perhaps space) — Ludwig V

I cannot fathom what you might mean. A sequence does not approach its limit in time. The limit of 1/2, 1/4, 1/8, ...is 0 right now and for all eternity. The fact is inherent in the axioms of set theory, along with the usual constructions and definitions of the real numbers and calculus. In that sense the fact "came into existence" when Newton thought about it, or maybe when Cauchy formalized it, and so forth.

But the history of our understanding of the fact is not the same as the fact itself. The earth went around the sun even before Copernicus had that clever idea. Likewise every convergent sequence always converged to its limit, independently of our discovery of those limits, and our understanding of what a limit is.

Is this your point of contention or concern? That you think that time is hiding in there somewhere? I profoundly disagree. You greatly misunderstand mathematics; or you have an interesting and original philosophy of mathematics; if you believe there's time hiding inside mathematics.

Perhaps they are all metaphors and there are different ways of expressing them that are not metaphorical. Is that the case? I recognize that I may be talking nonsense. — Ludwig V

If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.

Have I got that right?

Then you can't argue with me that you can argue with me. — TonesInDeepFreeze

Correct. Which is why I acknowledged your complaints and said nothing else. If I did, you'd complain that I was minimizing my apology by contextualizing it, either with snark or denial.

So I didn't even apologize. I acknowledge your complaints and I stand mute. I have nothing to say at all.

That is my message. It is on the "Infinite Staircase" thread, and does not include any of the passages attributed to me in your quotations. So I have no idea who wrote them. — Ludwig V

Oh I'm terribly sorry. That was for @keystone. I think I know what might have happened. Sometimes when I'm responding to someone, I have their quote tag in my copy buffer. Then I go to a different mention to reply to that, and when I was talking to keystone, I might have had your quote tag there instead.

No harm no foul I hope.

Did you try to make sense of my 'elevator pitch'? I wasn't communicating nonsense. It wasn't even my work I was talking about... — keystone

What did I miss?

Epsilon, eh? Will a few figures push you over the edge? That's where I need to go to move this forward... — keystone

I've done all I can. We're at a point where our interests have diverged. I need to wrap up my end so perhaps if you have any final questions, or perhaps if you come back now and then with more ideas formed over time, we can chat. So a pause, if nothing else.

Perhaps the paper by Milad Niqui. In that case things may get technical and out of the realm of TPF. — jgill

That's a bridge too far for me.

The objects that constitute both Euclidean and non-Euclidean (the unending many of them) spaces are abstract and both exist. Those objects may be applied in our scientific theories because a description of these objects can also describe some phenomenons in the real world. The problem is how do we get knowledge of these objects, if they are not physical? That is Benecerraf's problem. — Lionino

Yes I see what you meant. Thanks.

No I'm not. I accept that one of the premises of the thought experiment is physically impossible. That doesn't then mean that we cannot have another premise such as "there are no spontaneous, uncaused events".

You seem to think that because we allow for one physical impossibility then anything goes. That is not how thought experiments work.

It is physically impossible for me to push a button 10100100 times within one minute, but given the premises of the thought experiment it deductively follows that the lamp will be off after doing so. Your claim that the lamp can turn into a plate of spaghetti is incorrect. — Michael

I respectfully leave this conversation. We've said it all. i've enjoyed our chat.

Interesting. I hope I didn't bury the lede. I'm not all up about sarcasm. Rather, what I find important is (1) striving not to misrepresent a poster's remarks and to stand corrected when it is pointed out that one has; and (2) not to argue by ignoring key counter-arguments and explanations; not to just keep replying with the same argument as if the other guy hadn't just rebutted it. — TonesInDeepFreeze

I am so appreciative that you straightened me out on this extensionality thing that I can't argue with you about anything. I accept all your criticisms. You say I've done these things and I don't deny them. I make no defense nor explanation.

I do have a sarcasm gene and that rarely works online. You'd think I'd learn.

And to answer your question: No, I definitely do not have any interest in "picking fights" and I find no value in fighting for the sake of fighting. But I do find value in posting disagreements and corrections, whether regarding the math and philosophy or regarding the personal specifics of the posting interchanges. — TonesInDeepFreeze

Ok no more snark.

You first claimed that I was offensive to you. So I pointed out that you don't realize how offensive you often are. So I just gave you that info. I don't sweat being offended in posts. But you carelessly misconstrue what I've posted, and claim I've said things I haven't said, and write back criticism of my remarks by skipping their substance and exact points. And that is what I post my objections to.

Meanwhile, what you say about my posting style is rot. You say it's too long. But you also say it doesn't explain enough. Can't have it both ways. And I do explain a ton. But, again, I can't fully explain without having the prior context back to chapter 1 in a text already common in the discussion. And l explain somewhat technically because being very much less technical threatens being not accurate enough. Meanwhile, your own posts are usually plenty long, so take that tu quoque. — TonesInDeepFreeze

Are you just committed to picking fights with me? I've apologized several times tonight, for sins real and imagined. And some cosines too. Enough bro'.

My response was to 'what's wrong with you tonight?', not so much to 'wut?'.

Convenient for you now to self-justify by highlighting 'wut?' and not 'what's wrong with you tonight?'.

There was nothing wrong with what I posted that night. You just lashed out at as if there were, when actually the problem is that you, as often, reply to your careless mis-impression of what is written rather than to what is actually written. — TonesInDeepFreeze

Ah. The what is wrong with you and not the wut. I can see that now that you mention it.

I am terribly sorry to have offended you once again.

Does the axiom of identity mean Ludwig V = keystone ? — jgill

LOL I don't think so but I see what you mean.

This is yet another instance of you lashing out against something that I wrote without even giving it a moment of thought, let alone maybe to ask me to explain it more. Your Pavlovian instinct is to lash out at things that you've merely glanced upon without stopping to think that, hey, the other guy might not actually being saying the ridiculous thing you think he's saying. Instead, here you jump to the conclusion that "there's something wrong" with him. — TonesInDeepFreeze

I traced back to your mention of the axiom of infinity, and I still fail to see the relevance of the remark in context. I apologize for lashing out regardless. "wut" is a standard Internet location, and though it carries a bit of snarkitude, it's not considered overly aggressive in the scheme of things. Just an expression of puzzlement.

I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want? — TonesInDeepFreeze

Yes, point being that if I'm away from the board for a while I have no recollection of what threads o conversations I'm involved in. I look up my mentions and work through them. If I don't see a mention, I may miss your post.

I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts. — TonesInDeepFreeze

I would say that your communication style, with me at least, tends to be confusing. The only thing you wrote that made sense to me was the challenge to prove the transitivity of set equality. Once I realized I needed the converse of extensionality, I was enlightened.

Many other things you wrote were lost on me. I know this frustrates you, but it's like fishing. You had to go through a whole container of worms to finally hook the fish(fry). You should be happy, instead of complaining about the wasted worms.

I apologize for the typing things in remark. I must have written that before I understood your point.

Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.

It starts with these good posts: — TonesInDeepFreeze

What dissension? I'm happy I understood your point. I prefer not to go back into the old posts.

OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? — Metaphysician Undercover

In general, excepting the somewhat lesser-known example of set theories with urlements, the elements are other sets. If we are justified, given the axioms (whichever we choose) of set theory, to write:

$x \in y$

then we may colloquially read this as, "x is an element of y." That's what an element is.

It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. — Metaphysician Undercover

Actually I am wrong about that @TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now.

But what type of abstraction is it, one which we pretend is a particular? — Metaphysician Undercover

The law of identity applies to sets. So this line of argument is null and void.

Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals. — Metaphysician Undercover

If your criterion is that they satisfy the law of identity, they do. So your concern is addressed.

Isn't that exactly what meaning is, obeyance of some rules? — Metaphysician Undercover

Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets.

Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set. — Metaphysician Undercover

Typically it's another set. Sets are subject to the law of identity. This should satisfy your concerns.

It seems you are having problems understanding the inherent difficulty of the empty set. — Metaphysician Undercover

I believe in the field of psychology, this is known as projection. YOU have problems with the empty set. I have no such problems. The empty set is the set of purple flying elephants in my left pocket. Oh wait you don't like "real life" examples. Never mind.

The empty set is the set of things that violate the law of identity. In symbols:

$\emptyset = \{x : x \neq x \}$

Happy now? (Of course you're not!) There are other formulations.

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

I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set. — Metaphysician Undercover

An element of a set is a the left side of an expression $x \in y$ that can be deduced from the axioms of set theory. x is the element, and y is a set. But x is typically a set as well. Think paper bags inside of paper bags. Oops there I go with real world analogies again.

Yes, but you also claim that sets have no meaning. — Metaphysician Undercover

They can be viewed that way from a formalist perspective.

It's of no importance to set theory. Certainly sets don't necessarily have real-world referents, since sets are quite a bit stranger than paper bags or collections in general.

What of it?

An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory. — Metaphysician Undercover

What does chess mean?

See https://plato.stanford.edu/entries/abstract-objects/ and tell me if you find anything interesting in there.

Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal. — Metaphysician Undercover

Do you see the difference between the concept of set, and the concept of the set {1, 2, 3}?

One's a general set, and the other's a particular set.

Since you won't define a universal in such a way that you can sort this terminology out, I think your idea of universals must be vacuous. Fish is to this particular tuna on the end of my fishing line, as sets are to the set {1,2,3}. There is nothing problematic about that.

You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain? — Metaphysician Undercover

Me? I'm making no such suggestion.

Bijection is a problem, because it requires that the elements are individuals, particulars, — Metaphysician Undercover

They are, as far as I understand your use of the terminology, which you refuse to explain.

which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed. — Metaphysician Undercover

An element is a set in a set theory without urelements. We say x is an element of y if we can legally write $x \in y$. Nothing could be simpler.

Here is another real world example.

Fairy tale characters are an abstract universal. They are general, and they don't actually exist.

Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.

Fairy tale characters are abstract universals, and Cinderella is an abstract particular.

In your world you don't have any abstraction at all. I think you're taking a point too far.

Niqui arithmetic: Niqui's method allows you to take as input a symbol and a pair of locations in an unlabelled tree and it returns a corresponding location in the tree. It does not presuppose any mathematics other than Peano arithmetic. — keystone

I"m happy you find meaning in this.

My interpretation: It just so happens that if you label the nodes of the tree according to Stern-Brocot then those symbols correspond to the familiar operators of arithmetic.
Why this is important: If one can informally say that Peano defined the natural numbers according to discrete ordered positions along a line then that is no different than me saying that Niqui defined the fractions according to discrete ordered positions along a tree. — Ludwig V

I am happy this para has value for you.

I mentioned this only to suggest that my view may not be pre-axiomatic. I think Peano arithmetic is very important and Niqui took that to the next level. I also don't care to converse about proof assistants. — Ludwig V

That's a relief.

Sorry, I meant to say "an infinite number of natural numbers" as in "ℵ0
ℵ
0
natural numbers". I can see how this was misleading because when I later wrote "arbitrary natural numbers" I was referring to placeholders that can be populated by any natural number you come up with. — Ludwig V

Common error.

I'm just looking at things from the perspective of a computer. A computer doesn't access infinite sets, it always works with the finite set of finite inputs provided to it - so why not only assert the existence of those inputs (and whatever abstract objects it actually manipulates to deliver an output) and see how far this restricted math can go? — Ludwig V

Why not? I haven't told anyone not to do this.

What I'm proposing is not entirely philosophical. — Ludwig V

It's not math.

I think we got stuck in the weeds because I began to justify how fractions can exist in my view but that justification doesn't interest you. — Ludwig V

Correct.

I think temporary suspension of disbelief is probably the best path forward so that we can jump to the good stuff before you decide to quit...or have you already decided... — Ludwig V

I'm within epsilon. I no longer have any idea what we are conversing about.

Warning, Long-assed post ahead. Please tell me if I'm on target with your concerns.

I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though. — Ludwig V

The jokes illustrate the principle. The mathematicians takes the kettle off the stove and places it on the floor, reducing the problem to one that's already been solved.

It's called a performative speech act. Do you know about them? — Ludwig V

That tingled the circuit in my memory bank. Searle's doctoral advisor Austin talks about speech acts, and I believe Searle does too. That is everything I know about it. Not really clear what it's about.

Very roughly, the saying of certain words is the doing. The classic example is promising. A particularly important - and complicated - variety of speech act is a definition. Particularly interesting cases are the definition of rules. (Well, definitions are always regarded as rules, but there are cases that are a bit tricky.) — Ludwig V

Well I'm not sure I see what those examples are driving at. Speech where the speech is also an act. So, "It's raining out," is not a speech act, because I haven't done anything, I've only described an existing state of affairs. But telling you how the knight moves in chess (example of a rule] is a speech act, because I've brought the chess knight into existence by stating the rule. Something like that?

The relevance is that I'm puzzled about the relationship between defining a sequence such a "+1" and the problem of completion. — Ludwig V

It's very simple. First, by "+1" do you mean Peano successors? You used this notation several times in what follows and I am not sure I know exactly what you mean.

In Peano arithmetic (PA), we generate all the natural numbers with two rules:

* 0 is a number; and

* If n is a number, then Sn is a number, where S is the successor function.

We can use these two rules to define names like 1 = S0 and 2 = SS0 and so forth, and then use the successor function to define "+" so that we can prove 2 + 3 = 5 and so forth.

There is no "completion" of the sequence thereby generated, 0, 1, 2, 3, 4, ...

In particular, there is no container or set that holds all of them at once. The best we can do is say that there are always enough of them to do any problem that comes up in PA.

That gives you one logical system, PA, that has a certain amount of expressive power. We can do a fair amount of number theory in PA. We can NOT do calculus, define the real numbers, define limits, and so forth.

In PA we have each of the numbers 0, 1, 2, 3, ... but we do not have a set of them. In fact we don't even have the notion of set.

Next step up is set theory, for example ZF, that includes the axiom of infinity. The axiom of infinity actually defines what we mean by a successor function for sets; and says that there is a set that contains the empty set, and if it contains any set X, it also contains the successor of X.

This gives you something PA doesn't: A "container that holds all of 0, 1, 2, 3, ... at once, in fact not just a container, but a set, an object that satisfies all the other axioms of ZF.

We can then show that the axiom of infinity lets us construct a model of PA within ZF; and we take that model to be the natural numbers.

The tl;dr is this:

PA gives you each of 0, 1, 2, 3, ...

ZF with the axiom of infinity gives you {0, 1, 2, 3, ...}; that is, all the marbles AND a bag to put them in.

Hope that wasn't too much information, but it's the way to think of "potential" versus "completed" infinities, which are philosophical terms that don't really find use in math.

Each element of the sequence is defined. Done. (And an infinite number of tasks completed, it seems to me). — Ludwig V

Mathematical sequences and supertasks are two entirely different, but strongly related, ideas.

There is no time in mathematics. But supertasks are all about time. That's where a lot of the confusion comes in. Supertask discussions talk about time, which is a physical concept; but then examples like Thomson's lamp posit circuits that can change state in arbitrarily short intervals of time, which is a decidedly NON-physical idea. It's a fairy tail (under currently known physics). That's where much of the confusion comes in.

So I hope that you can separate out these two concepts. Are you asking about mathematical sequences, such as 1/2, 1/4, 1/8, ... that have the limit 0? That is a completely understood subject in math.

Or are you imagining that someone "speaks these fractions out loud" in their corresponding amount of time, thereby "saying them all in finite time?" This is a totally nebulous, made-up conceptual fairy tail that is the cause of much confused thinking among philosophers.

But apparently not dusted, because we then realize that we cannot write down all the elements of the sequence. — Ludwig V

This is actually not much of an objection. It is far too weak. We cannot write out all the terms of any sufficiently large FINITE sequence, either. You can't write out the numerals from 1 to googolplex in y you lifetime at one number per second. It would take longer than the age of the universe.

So you are not making any substantive objection.

In PA the numbers are conceptually created one at a time, but they're really not, because there is no time. 0 is a number and S0 is a number and SS0 is a number, "all at once." You can call that completion if you like.

In ZF, it's more clear. There is a set that contains 0, 1, 2, 3, ... You can give the set a name and you can work with it.

But either way, your concept of completion involves time; and as I've noticed, that involves CONFUSING mathematical sequences, about which we have perfect logical clarity; with supertasks, about which we have much pretentious confusion.

In addition to the rule, there is a distinct action - applying the rule. That is where, I think, all the difficulties about infinity arise. — Ludwig V

No, that is something you are bringing to the table, but that I don't think is correct. There's no distinct action of applying the rules.

In the PA incantation: 0 is a number and Sn is a number if n is; that creates all the numbers. There is no time involved. Time is a factor that you are letting confuse you.

We understand how to apply the rule in finite situations. But not in infinite situations. — Ludwig V

We understand how to apply successors perfectly well in the infinite situation. In fact the rule that "If n is a number, then Sn is a number," is an instance of induction, or its close relative, recursion. These things are perfectly well understood.

Think of applying "countable" or "limit" to "+1". The concept has to be refined for that context, which, we could say, was not covered (envisaged) for the original concept. — Ludwig V

You are making this up out of some level of confusion involving time. Time is not a consideration or thing in mathematics. All mathematics happens "right here and now."

I am trying, I don't know if I'm getting through or not, but I am trying to get you to separate out your naive notion of timeliness in mathematics, with mathematics. Time matters in physics and in supertask discussions. It's important to distinguish these related but different concepts in your mind.

(By the way, does "bound" in this context mean the same as "limit"? If not, what is the difference?) — Ludwig V

Good question. A bound and a limit are two different things. A couple of examples:

* Consider the set {1/2, 1/4, 1/8, ...}.

-43, that is negative 43, is a lower bound of the set. 0 is the "greatest lower bound," a concept of great importance in calculus.

* But here's a more interesting example. Consider the sequence 1/2, 100, 1/4, 100, 1/8, 100 ...

It has two limit points, 0 and 100. But it has no limit, because the formal definition of a limit is not satisfied. To be a limit the sequence has to not only GET close to its limit, but also STAY close.

0 and 100 would be the greatest lower bound and the least upper bound, respectively.

Now I know this was too much info!! This is just technical jargon in the math biz, don't worry about it two much. But bounds and limits are different concepts. Limits are more strict.

Oh, yes, I get it. I think. — Ludwig V

Maybe that bit about the order topology was a little too much. My only point is that there is a mathematical context in which omega as the limits of the natural numbers is the same as calculus limits. That's all I need to say about that.

Forgive me for my obstinacy, but let me try to explain why I keep going on about it. I regard it as an adapted and extended use of the concept in a new context. (But there are other ways of describing this situation which may be more appropriate.) — Ludwig V

This didn't parse, I don't know what you are referring to. What is "it" and "this situation." Nevermind I'll work with the rest of the text.

My difficulties arise from another use of the "1" when we define the converging sequence between 0 and 1. It seems that there must be a connection between the two uses and that this may mean that the sense of "limit" here is different from the sense of ω in its context. In particular, there may be limitations or complications in the sense of "arbitrary" in this context. — Ludwig V

This is a little convoluted and confused. What converging sequence between 0 and 1? Say we have the sequence 1/2, 1/4, 1/8, ... for definiteness.

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs $\frac{1}{2^n}$. I'm starting from 1 rather than 0 for convenience of notation, it doesn't matter.

Now in order to formalize where the limit 0 fits into the scheme of things, we can say that the limit is the value of that function at the point $\omega$ in the EXTENDED natural numbers

0, 1, 2, 3, ...; $\omega$

Those are NOT the natural numbers. I've stuck a conceptual "point at infinity" at the end. I hope this is not confusing you. Tell me what your concerns are.

I thought so. So when the time runs out, the sequence does not? Perhaps the limit is 42. — Ludwig V

The "termination state" is 42. 42 is not the limit of the sequence 0, 1, 0, 1, ... The word limit has a very technical meaning. It's clear that the sequence does not "get near and stay near" 42.

That's why for purposes of analyzing supertasks I am DEFINING the phrase "termination state" of a sequence to be a value "stuck at the end," but that is NOT NECESSARILY A LIMIT.

I hope this is clear. The termination point is arbitrary, it can be 42 or a pumpkin. But in no case are those values limits in the calculus sense.

So we say that all limited infinite sequences converge on their limits. — Ludwig V

Hmmm. "Limited" is not a term of art in this context. Given a sequence, it either converges to a limit or it doesn't. A convergent sequence of course converges to its limit, but this is a tautology that follows from the definition of convergence to a limit. A convergent sequence converges to its limit, but that doesn't really any anything we didn't already know.

Believe it or not, that makes sense to me. Since it is also an element of the sequence, it makes sense not to call it a limit. — Ludwig V

Glad it makes sense, but the limit is NOT repeat NOT part of the sequence.

It's part of what I'm calling the extended sequence, with the limit or termination point stuck at the end. But that is my terminology that I am making up just for these supertask problems.

Hope that's clear.

When I write my semicolon notation: 1/2, 1/4, 1/8, ...; 0

that is a fishfry-defined extended sequence. The sequence is 1/2, 1/4, 1/8, ..., and the limit is 0.

I use this notation to describe the termination state of a supertask: on, off, on, off, ...; pumpkin

The sequence is the on/off part; the pumpkin is the termination state.

Hope this is getting clearer.

I have completist tendencies. I try to resist them, but often fail. — Ludwig V

I don't even know what that means :-) What are completist tendencies?

You would hate the rational numbers then. They are not complete. For example the sequence 1, 1.4, 1.41, 1.412, ... where each term is the next truncation of sqrt(2), does not have a completion in the rationals.

The real numbers are the completion of all the sequences of rationals. That's how we conceptualize the reals.

Well I wrote a lot, let me know if any of this was helpful and let me know what's still troubling you.

tl;dr to this entire post:

Mathematical sequences are clear and rigorous. We have a fully worked out theory of them.

Supertasks are nebulous and vague. Reason: There is no time in math. Time is a concept of physics. And Supertask problems always involve physical impossibilities, like flipping a lamp in arbitrarily small intervals of time. That's the source of all the confusion. Supertasks are fairy tales, like Cinderella's coach; and you can no more apply logic to a supertask problem than you can to the coach turning into a pumpkin.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part. — ssu

IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course.

I don't understand your question. — Michael

Ok.

Asking me why I'm using P1 as a premise is as nonsensical as asking me why I'm using P2 as a premise. They are just the premises of the thought experiment. The intention is to not allow for the lamp to be off, for the button to be pushed just once, turning the lamp on – and then for the lamp to be off. — Michael

Pending either of us having anything new to say, I am out of this conversation.

We are trying to understand what it means to perform a supertask, and so we must assert that nothing other than the supertask occurs. There are no spontaneous, uncaused events. If we cannot make sense of what the performance of the supertask (and only the supertask) causes to happen to the lamp then we must accept that the supertask is metaphysically impossible. — Michael

"We" does not include me. I regard Thomson's lamp as a solved problem. When you say "there are no spontaneous, uncaused events," you are ignoring the physically impossible premises of the problem. Pushing a button in an arbitrarily small time interval to activate a circuit that likewise switches in an arbitrarily small time interval is already a spontaneous, uncaused event. That's why I commend to you the parable of Cinderella's coach.

What I do not like about the SCOTUS ruling - I read it (quickly) - is that they appear to have completely sidestepped common sense — tim wood

For what it's worth ... liberal legal scholar Jonathan Turley -- although liberals have disowned him, now that he turns out to be off the reservation a bit -- has an article out in the (scurrilous right wing rag) New York Post titled,

Supreme Court’s Trump immunity ruling is what the body was designed for — unpopular but constitutionally correct

https://nypost.com/2024/07/01/opinion/supreme-courts-trump-immunity-ruling-is-what-the-body-was-meant-for-unpopular-but-constitutionally-correct/

It's worth a read if one is openminded about these kinds of things. The Supes didn't give presidents blanket immunity. They said that what counts as "official acts" are to be determined by the lower courts. So they're not issuing an edict. They're leaving everything open for the judicial process to play out. If you take the long view -- and that's the whole idea of lifetime appointments, the Supremes are not supposed to be buffeted about by the latest momentary passions of society, that's what the every-two-years House of Representatives is for -- so if you take the long view, this is somewhat of a moderate, sensible decision. After all, Joe Biden is president right now, and he has all the freedoms granted under this ruling. Which are none at all. The president's actions are always subject to judicial review as to whether they're official acts.

So I think this is kind of sensible, political howling and screeching from the left notwithstanding. The left calls the court illegitimate every time the court does something they don't like. In fact just last month Biden was telling us how wrong it was to criticize a legal decision when a case went against the right and the right was howling. It's all politics and this is an especially volatile election year.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me. — TonesInDeepFreeze

If I crossed any lines, I apologize. But I think you are equivocating the word "insult." If I tell you, Tones, you are a low down rotten varmint who cheats at cribbage!" that's an insult.

But if I don't happen to dwell on every word you write; and if I often find your expository prose convoluted and unclear, especially when you lay out long strings of symbols without any context; my eyes do glaze over, and I do skip things.

That is not an insult. It's just me being me, reacting to whatever you wrote that made my eyes glaze. The fault is all mine, But that's who I am and how I am. I am not insulting you.

Can you see the difference between:

(a) Me actively and directly insulting you; and

(b) Me just being my highly imperfect self, doing something that annoys you.

Surely you can see the difference.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced. — TonesInDeepFreeze

Well that's good, so let's go with the grace.

I am hopelessly behind composing posts in at least a few threads. Even years behind in threads that I just had to let go because I really should be spending my time on other things more important than posting. — TonesInDeepFreeze

Shouldn't we all!

EDIT: After posting this I realized that there might be some confusion about Niqui Arithmetic. I have since posted another message entitled NIQUI ARITHMETIC. Please read that first. — keystone

I will dispatch a clone. Meaning, I will add it to my input queue, which is now very long. If you supply a two-sentence summary I'll read it. In Silicon Valley they call it your "elevator pitch."

Peano arithmetic can be formalized in Coq. — keystone

Oh now I have to converse about proof assistants? You know, if you've been picking up the lingo, that's great. Not of interest to me. It's impressive what they're doing. Just not an interest of mine.

Similarly, Niqui arithmetic on the SB tree, which builds on Peano arithmetic, has been proven in Coq. There's an unquestionable structure to natural numbers and fractions that we both agree on. What we disagree on is the ontology related to these necessary truths. You believe that Peano arithmetic applies to infinite natural numbers, — keystone

No such thing as infinite natural numbers, nor do I hold any such belief, mental state, or psychological disposition towards any such thing as what you wrote.

whereas I believe it applies to arbitrary natural numbers. By this, I mean that Peano arithmetic corresponds to an algorithm designed to take as input any arbitrary pair of natural numbers and output the expected natural number. My ontology does not require the existence of any number. I only need numbers when I want to execute the algorithm, and I only need two numbers at that, not an infinite set. — keystone

I think that's great. I have no response. I have no function here except as a sounding board. Honestly you sound very crankish about all this. Why not just go learn some math.

Although the above focuses on Peano arithmetic, the same applies to Niqui arithmetic. While the actual computations of Niqui arithmetic involve the manipulation of symbols or electrical signals, an elusive structure emerges in our mind when studying the algorithm—the SB-tree. Nobody has ever envisioned the complete tree, but we have seen the top part, and when I say 1/1 occupies a particular node, that top part is all I need to see. I don't need to assert the existence of an unseen complete tree; after all, it is merely an illusion that helps us understand the underlying algorithm (Niqui arithmetic). — keystone

\

I'm glad you find meaning in this.

I'm trying to establish parallel ontologies: Actual vs. real. At this point, we have actual numbers (fractions) and real numbers. We have actual points (k-points corresponding to fractions) and real points (k-lines corresponding to real numbers). This distincting is rather bland in 1D but it becomes much more consequential in 2D when establishing a foundational framework for geometry and calculus. — keystone

I'm not the guy for this any longer.

The Philosophy Forum appears to be quirky. I tried quoting this multiple times, sometimes including the spaces surrounding it, sometimes not, and about half the time it puts a column of 1-character lines. — keystone

The Philo forum giveth, and the Philo forum taketh away. I have learned that over the years.

As per P1, the lamp cannot spontaneously and without cause turn into a pumpkin, — Michael

Question: Do you put the same constraint on Cinderella's coach? Why or why not? Want to understand your answer.

Regarding the rest of it, I'm lamped out, so I will not debate your ideas further. We have heard each other's talking points multiple times at this point. At least I've got a big time philosopher on my side. How cool is that, right? To actually have professional vindication for a personal idea. I've gotten more than my money's worth from this conversation.

If you feel like answering whether you put the same constraint on Cinderella's coach, I'd be intereted to know. Can't respond anymore to the rest of it. When I get to the point that I haven't typed any words on the subject that I haven't typed before, that's how I know I"m done with that topic.

Thanks for the chat plus any Cinderella comments.

It seems that from you I get extremely good answers. — ssu

Thank you.

Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this. — ssu

If you're interested in this stuff, do you know the nLab Cafe? It's a category theory wiki. Here's their page on the theorem

It's all very categorical. Like a new paradigm for thinking about math.

And that not necessary is important for me. This is what TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this. — ssu

I'm not sure how the subject came up. It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing.

Quite so. Except I thought that it had actually been done. — Ludwig V

Use of language. When a mathematician says, "X can be done," that's just as good as doing it. There are many jokes around that idea.

There's a formalism or concept called the order topology, in which you can put a topological structure on the set 0, 1, 2, 3, ..., $\omega$ such that $\omega$ is a limit point of the sequence, in exactly the same way that 1 is the limit of 1/2, 3/4, 7/8, ...

A topological structure is an abstraction of expressing closeness with open intervals, as in the real numbers. The point is that $\omega$ is the limit of 0, 1, 2, 3, ... in exactly the same sense as "abstracted freshman calculus," if you think of it that way.

Quite so. That's why I specified "convergent sequences". (I don't know what the adjective is for sequences like "+1" or I would have included them, because they also have a limit.) "0, 1, ..." is neither. Does the sequent 0, 1, ... have a limit - perhaps the ωth entry? — Ludwig V

No. 0, 1, 0, 1, ... does not have any limit at all. And we can even prove that. Note that it has two subsequences, 0, 0, 0, ... and 1, 1, 1, ,,, that each have respective limits of 0 and 1.

Now it's a theorem that if a sequence converges, all of its subsequences must converge to the same limit. Makes sense, right? A convergent sequence "squishes down" to near the limit.

So a sequence like 0,1,0,1 ... that has two subsequences with different limits, proves that the sequence can not have a limit.

Also, I don't think there even is a name for an arbitrary termination value for a non-convergent infinite sequence. Like

1/2, 3/4, 7/8, ...; 47

In this case 47 is still the value of the "extended sequence" function at $\omega$. I call it the terminal state.

I've never seen anyone else use this idea as an example or thing of interest. It doesn't have a name. But to me, it's the perfect way to think about supertasks. The terminal state may or may not be the limit of the sequence; but it's still of interest. It could be a lamp, or a pumpkin, or it could "disappear in a puff of smoke."

If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time. — TonesInDeepFreeze

Not link, quote. Either quote a fragment of my post, as I just did to yours; or else just mention me as @fishfry, where you have to type "" around the handle name.

Linking posts is something else, at the bottom you can get a hard link to the post, but you don't need to do that.

My preference regarding you is that you don't gloss my posts and jump to conclusions that I've said something I didn't say but that you think I must have said in you own confusions or lack of familiarity with the concepts or terminology. — TonesInDeepFreeze

Something about the scorpion and the frog. You expect me to stop having the many flaws I have? I will do my best, but interact with me at your discretion.

I have a large pile of mentions, so I'll get to them and save yours for later. I'm still in the afterglow of my set theoretic epiphany. I understood your point. You're right and I was confused, but now, thanks to your untiring efforts, you have unconfused me. Actually I think I was just hallucinating, because I do know that extensionality is an implication and not a bi-implication. I just never thought about the converse. But the converse is the "portal to the next level down," predicate logic.

I'm happy to have clarified this, it makes a lot of sense.

I think we can jump forward past the extensionality. The moment I saw the problem with proving the transitivity of set equality, I was enlightened. I swear, I almost literally smacked my head. "I can't use extensionality. I need the converse. So you picked the perfect puzzle to get through to me.

So going forward, I stand educated on this point. And although I do try my best not to exhibit my flaws, well, I may yet leap to an unwarranted conclusion now and then.

Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and communication have won the day finally. — TonesInDeepFreeze

Yes. Quite the epiphany. I've actually just found several web pages and articles explaining all this. One even mentioned that the converse of extensionality follows from Leibniz (either ident of indisc. or other way 'round). Evidently I'm the last person to find this out. Even the Wiki page on extensionality mentions this, and I thought I'd read it several times but evidently not that part.

Late here way past bedtime I actually need to be somewhere tomorrow morning I'm going to regret this. Will be offline till tomorrow evening or day after.. Thanks for the insight. It was the proof of the transitive property that did it. Once I realized I needed the converse, the floodgates opened. Great example.

Ok more later. Thanks again.

@TonesInDeepFreeze

I didn't get to your most recent yet. But I did have a bit of an epiphany and it's possible you may be steering me to enlightenment.

I started working on (x = y & y = z) -> x = z, which seems an easy consequence of extensionality.

So I started by writing down the statement of extensionality, and right away I see that I'm in trouble! I don't need extensionality ... I need the converse of extensionality. I need to go from x = y to saying that for all z, z in x iff z in y. That is not given by the axiom. So you have actually taught me something.

I did a quick search, and found this: The converse of the axiom of extensionality where he says that the converse "follows from the substitution property of equality."

So this is quite a bit more subtle than I thought, and I will have to work on this some more. I do think you have made your point, at least provisionally. I can't assume the converse of extensionality. Who knew, right?

In which case ... an axiom is NOT the same as a definition, because definitions are reversible, and extensionality is not. So I do believe you may have made a couple of good points with this example.

ps -- Ah ... the Wiki page on extensionality explains your point that if equality is not a primitive symbol in predicate logic, then extensionality is taken as a definition rather than an axiom. You did say this to me several times. I now begin to see your point.

This example did it for me. I have to study this a bit. I did not realize that extensionality goes only in one direction, and that the = symbol is not being defined, but is inherited from the underlying predicate logic. You have made your point. I need to understand this better.

That is where he gives the semantics for '=', as I mentioned that '=' is given a fixed interpretation. — TonesInDeepFreeze

Ok I'll check again. I'm reading the SEP article on identity, and it's interesting reading. Puts some of what you've been saying in context. They did say that "Leibniz’s Law must be clearly distinguished from the substitutivity principle ..." so perhaps that's pushback to your claim.

But there are actually two principles, identity of indiscernibles and indiscernibilility of iten ...

oh man i'm typing and you are replying back. I can't keep up. Let me just say that the SEP article is interesting and I'll get to the rest of this tomorrow.

However! You just said

So in set theory Ax x=x is redundant. — TonesInDeepFreeze

in which case you agree with my main point and there is nothing more to say. That's why I'm confused. Once you concede that identity is not necessary for set theory, then I don't know why you are going on about set theory.

(x = y & y = z) -> x = z — TonesInDeepFreeze

Challenge accepted, I will get to this tomorrow or day after, I am a little busy tomorrow.

So I appreciate that you are now writing much shorter posts, making it possible for me to read them. But you are compensating by writing very quickly, so that right now I'm two posts behind and you have one or two already ahead of me! I can't keep up.

So let me work on (x = y & y = z) -> x = z and I'll read through your posts tomorrow. It's after midnight right now where I live.

I reply to your posts, then I see your replies back, to which I then reply back ...

I don't use any other protocols. — TonesInDeepFreeze

That only works if either (a) I happen to read all the recent posts in a given thread, which I rarely do; or (b) I happen to be posting at the same time as you.

I'm sure you can see this leaves a window where I might not see your posts.

And you can look at the SEP article 'Identity' where you'll see:

Leibniz’s Law, the principle of the indiscernibility of identicals, that if x is identical with y then everything true of x is true of y. — TonesInDeepFreeze

Ok thank you for that specific reference. You should know that I generally respond to my mentions and don't always monitor the threads. Please give me a mention when you want me to see your posts. Of course that doesn't guarantee I'll see everything you want me too, but at least I'll know you said something to me.

I hope you know that 'the crank' does not refer to you. If that was not clear in the context, then I should have made it clear. — TonesInDeepFreeze

Oh thanks. I dropped by the site and saw I had 6 mentions and that they were all from you so I was snapping back pretty quickly without actually reading much.

So I saw a ref to equality on 112 of enderton that had nothing to do with set theory, and can't find anything at all on page 83. But on 112 he said that we can take as a rule x = x. But we don't need that for set theory! This is my point, or point of confusion. If I want to know if x = x for some set x, I can just apply extensionality and check to see if if for all z, z in x iff z in x. Which is of course true. So x = x. I don't need the law of identity to determine if x = x if x is a set. This is my point.

I'm giving you a lot of the same information and explanation over and over, since you skip over it over and over. — TonesInDeepFreeze

It gets lost in all the verbiage and symbology.

I said that the indiscernibility of identicals is formalized in identity theory. I didn't say that any particular formalization mentions it with the phrase 'the indiscernibility of identicals'. — TonesInDeepFreeze

Ok. But you say this isn't written down anywhere?

The principle was enunciated by Leibniz. But in mathematics, it's often called 'the principle of substitution of equals for equals'. — TonesInDeepFreeze

This pushed hard against my understanding. The identity of indiscernibles says (afaik) that two things are the same if they share all properties.

Substitution says you can plug in things that are equal in expressions. I'm not sure how that relates.

And in modern logic, it is an axiom schema in the manner I've posted, which is equivalent (though notation and details differ) to Enderton and Shoenfield, for example. — TonesInDeepFreeze

I suppose I'll have to take your word for it, because it's not in Enderton or I missed your explanation earlier. I just glanced at the SEP entry for ident of indisc. and it doesn't say anything about substitution. If there's no written reference, is this perhaps an idea of your own?

In any event, if I want to know if two sets are equal I apply extensionality. And that's another reason I get lost in your posts. I don't see your point. Extensionality tells you everything you need to know about when two sets are equal. You don't need anything else.

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me. — TonesInDeepFreeze

ok

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer. — TonesInDeepFreeze

You're alternately insulting and praising me. Make up your mind!

Exactly. — TonesInDeepFreeze

Stop agreeing with me, that's no fun!

(edit) So you see I do know some logic after all!