I am not informed in physics. So anything I say about axiomatizing physics might very well need to be corrected or qualified. However, I can at least address the question as to basic logic.

Just for a minimal start, it is clearly evident that it is not in principle inconsistent merely to add non-mathematical axioms to ZFC, even if the non-mathematical axioms include a declaration that the non-mathematical domain is finite.

For the most simple example:

To ZFC, add a 1-place predicate symbol: P (intuitively, Px means x is a particle). Add an axiom E!x Px. Define: p = the unique x such Px. So the set of particles is finite.

Or, instead of having the axiom E!xPx, we could have: ExPx & ~Ey(y is infinite & Axey Px). Then there are an indeterminately many particles but not infinitely many of them.

[EDIT: The above two paragraphs are not what I meant:

Add a 1-place predicate symbol: R (intuitively, Rx means x is a particle). Add an axiom E!xAy(yex <-> Ry). Define P = {y | Ry}. Add axiom: P is finite.

And delete the second paragraph.]

Those are not inconsistent theories. [EDIT: That is not an inconsistent theory.]

In Z set theories, we may define:

x is a class <-> (x = the-empty-set or Ey yex)

x is a set <-> (x is a class & Ey xey)

x is a proper class <-> (x is a class & ~ x is a set)

x is an urelement <-> ~ x is a class

Then we have theorems:

Ax x is a class

Ax x is a set

~Ex x is a proper class

~Ex x is an urelement

That entails that every particle is a set. And we might not like that. But we could say "Oh well, that is an artifact of abstraction that won't hurt the physical theorems we'll prove, similar to the fact that the set theoretic definition of numbers burdens numbers with abstract set theoretical artifacts that however don't interfere with the mathematics we will do with those numbers."

Or we could reformulate as follows:

Delete the axiom of extensionality.

Add primitive: S (Sx intuitively meaning "x is a set").

Add axiom: E!x(Sx & Ay ~yex) (there is a unique empty set).

Add axiom Axy((Sx & Sy) -> (Az(zex <-> zey) -> x=y)) (revised axiom of extensionality).

Add axiom: Px -> x is an urelement. [EDIT: Rx -> x is an urelement]

Adopt all the other axioms of ZFC.

It seems a safe bet that that is a consistent theory. Granted, it is not ZFC and not strictly speaking set or even class theory (such as NBG) that both are characterized by perhaps the most crucial property of classes and sets - extensionality. But set theory with urelements may be recognized as a reasonable variant.

And one may add additional primitives such as a function for mass, primitives about spacetime, primitives and definitions about subparticles, physical objects made from particles, and axioms about particles, subparticles, and physical objects, their masses, interactions among them in space and time, etc... And with axioms with mathematics about the physical objects, etc, hopefully deriving theorems of physics.

I don't see that there would be a correct argument that merely in principle such a theory must be inconsistent.

/

See Suppes's 'Introduction To Logic' pgs 291-305 for another example: axioms for Particle Mechanics.

http://tph.tuwien.ac.at/~svozil/publ/1995-set.pdf

https://www.sciencedirect.com/science/article/abs/pii/S0960077996000550

and from a set theorist:

http://logic.harvard.edu/EFI_Magidor.pdf

I don't know your personal use of the terminology. Of course even without AC every finite set of nonempty sets has a choice function. I don't know whether LEM is needed for that (at least on first glance I don't see that it is).

I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite.
— TonesInDeepFreeze

Of course that is vacuously true, since there is no axiomatic formulation of physics. — fishfry

I thought it was obvious that since we don't have in front of us an axiomatization, then my question is hypothetical regarding whatever proposed axiomatization might be presented. Such an axiomatization would have mathematical axioms and also extra primitives and axioms for physics. I don't see that such a theory would have to be inconsistent. For example, "Ex x is infinite" is not inconsistent with "~Ex x has infinitely many particles" (I'm just using 'particles' as a placeholder for whatever technical rubric would actually be used).

The axiom of infinity is inconsistent with known physics since there is no principle of modern physics that stipulates the existence of any infinite set, — fishfry

That doesn't entail inconsistency. Just because a theory doesn't have a certain principle doesn't entail that adding that principle causes inconsistency. But if physics had a principle that it is not the case that there exists an infinite set, then yes, there would be inconsistency. But even if physics had a principle that there are not infinitely many particles, that is not itself inconsistent with the existence of infinite sets, such as infinite sets of numbers if numbers are not axiomatized to be particles.

contemporary physics can not accept the axiom of infinity as a physical principle. — fishfry

I never said that it would be a physical principle. It would be a mathematical theorem to which are added primitives and axioms for theorems of physics.

I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides.
— TonesInDeepFreeze

Why not? — fishfry

If I wanted to take a normative stance on the subject then I would.

why not say something like, "The axiom of infinity is a formal statement that, as far as we know, is false about the world, yet taken as a fundamental truth in mathematics. And I account for that philosophically as follows: _______." — fishfry

Because I am not motivated to do that. Moreover, I am not inclined to accept that my thoughts (including lack of conclusions) about a subject needs to conform to a Procrustean format "this and that, and I account for that philosophically ______."

Ducking the question doesn't help. — fishfry

I haven't made claims that I've ducked supporting.

The axiom of infinity is taken as true in "some abstract mathematical sense." My point exactly, on which we are now in agreement. There are models of set theory in which it's true; at least if there are any models at all. — fishfry

Of course, a consistent theory has models. But nothing I've said then commits me to adopting a position for or against the proposition that mathematical statements are true only relative to models.

You've come to be in agreement with me. The only way the axiom of infinity can possibly be accepted as true or meaningful is in the context of purely abstract math — fishfry

Nothing I've said commits me to such a claim. I said that I have a sense of the abstract meaning of mathematical statements (also, I can add, that certain axioms fit my intuitive concept of what sets are as abstractions). That doesn't entail that I also must go on to say that my sense previously described must be the ONLY correct, meaningful or useful sense, and especially it does not require that I take a stance that the notion of model-theoretic truth is the ONLY correct, meaningful or useful sense.

and NOT physics — fishfry

Again, I haven't claimed that the axiom of infinity "says" anything about physical objects or has anything to do with physics other than it provides the mathematics used for the sciences such as physics.

when we got into the second half of Halmos Naive Set Theory most of us lost interest. — jgill

And I have not the least interest in the subject of corporate financing. Go figure. But I do make it a point to go into discussion threads about business and economics and make my boredom with the subject well known.

If an inductive set that's not physical "exists," what does that mean to you? — fishfry

I'm not claiming any particular sense of existence. Nor am I disputing any particular sense of existence. In context of the question whether set theory is inconsistent with physics, I am interested in the context of formal axiomatization (I'll just say 'axiomatization'). In Z we have the theorem: Ex x is infinite. I would think that that would provide an inconsistent axiomatization T of mathematics/physics only if T has a theorem: ~Ex x is infinite.

If you play Platonist — fishfry

I don't.

But you're the one claiming that an inductive set "exists," — fishfry

I claim that set theory has a theorem: Ex x is an inductive set. I don't opine as to what particular sense we should say that provides. I do tend to think that whatever that sense is, it is at least some abstract mathematical sense. And I appreciate that there are variations held by different people. I can "picture" in my mind certain notions such as "the least inductive set is an abstract mathematical object that I can hold in my mind as "picked out" by the predicate of being a least inductive set". I find it to be a coherent thought for myself. But I don't have any need to convince anyone else that such a view of mathematical existence should be be generally adopted or even considered coherent by others.

"I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically" broke my parser. And that hurts! — fishfry

Then your parser is weak handling double negation. I chose double negation because it best suits the flow of how I think about the proposition. With less negation: I explained why previously that it was reasonable for me not to infer that you were writing hyperbolically.

If I say I am not replying, that would constitute a reply. — fishfry

Just to be clear, my replies were not merely to you saying that you are not replying.

inane self-referential conversation — fishfry

My part is not inane. And whether or not you think that conversation about conversation should be eschewed, I don't think that way.

When you something substantive, as opposed to looping back on the syntax of whatever I may have said — fishfry

I did not merely "loop back on the syntax of whatever you may have said". It's interesting that you want an end to posting about the conversational roles themselves, but you want to do that while still getting in your own digs such as "inane" and dismissive mischaracterization such as "looping back on the syntax".

The axiom of infinity is in contradiction with known physics — fishfry

I don't know that. The axiom of infinity says there is an inductive set and, with other axioms, entails that there is an infinite set. Set theory doesn't say that there is an infinite set of particles or that physical space extends infinitely outward or whatever. Also, is it definitively established that there are not infinitely many particles or that space does not extend infinitely outward?

You would not have begrudged me then, but you will begrudge the living bejeebus out of me now? — fishfry

No. I did not begrudge you hyperbole. Rather, (1) I explained why previously it was not unreasonable for me not to infer that you were writing hyperbolically, and (2) That even factoring for hyperbole, I disagreed with the non-hyperbolic claim behind the hyperbole.

Does the phrase, "Give it a rest," have any resonance with you? — fishfry

I don't see that when you continue to reply, with both repeated points and arguments and new points and arguments that I should then not also reply.

I don't remember making any recent argument with you other than that it's pointless to argue about how many mathematicians believe this or that — fishfry

The arguments are in your posts recently made.

I may be lost by now. I have no idea what we're discussing. — fishfry

You quoted me:

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness. — fishfry

And to that you replied that you were being hyperbolic (presumably when you said that nobody takes as axioms as true other than model-theoretically). So I replied that your hyperbole is not connected to what you quoted by me.

Do you mean the hyperboles "blow up the moon" and "AIDS denier"?
— TonesInDeepFreeze

No, those are literal facts of record. I supplied the relevant Wiki links. — fishfry

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Since the context was outlandishness, when you replied that you were being hyperbolic, my reply was wondering whether you meant that you were being hyperbolic when you compared "blow up the moon" and "no AIDS exists" to a view that mathematical truth is not limited to the model-theoretic. So, since that is not what you meant, now again, I don't know what connection you intended between my point about outlandishness and your having been hyperbolic in saying that nobody regards axioms as true except in the model-theoretic sense.

No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all."
— TonesInDeepFreeze

LOL. You said, "I don't know that it hasn't been axiomatized AT ALL," your caps. Which could mean:

a) You don't know AT ALL if it hasn't been axiomatized; or

b) You don't know if it's been axiomatized AT ALL, as opposed to in its entirety. — fishfry

I just explained very clearly that I meant that I don't know that physics has not been axiomatized to any extent. I'll say it yet another way: I don't know that there is no extent to which physics has been axiomatized.

(1) YOU said, "Physics has not been axiomatized at all". So I replied "I don't know that physics has not been axiomatized at all".

Obviously I'm replying to YOUR OWN sense of YOUR OWN statement "Physics has not been axiomatized at all". Especially in context, I didn't take YOUR OWN statement as ambiguous.

I pointed out that parts of science have been very nicely axiomatized, such as Newtonian gravity. That would be an agreement with (b). Whereas (a) refers to the state of your knowledge. — fishfry

You didn't mention Newtonian gravity in the post to which I first replied.

Here is the exchange:

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.
— TonesInDeepFreeze

This could never be true. Physics has not been axiomatized at all. They can't even reconcile quantum mechanics and relativity. And the idea that set theory could ever be a foundation for physics seems to me to be an unlikely stretch. But at least that is an interesting and substantive topic in the philosophy of math and science. — fishfry

Especially in context, I took you to mean that there has been no progress in axiomatizing physics.

If someone says, "The house has not been built at all", then one would understand that to mean that there has been no progress in building the house, that no one has completed even the first phase in construction of the house.

So when you said, "Physics has not been axiomatized at all", I took you to mean the obvious sense that there has been no progress in axiomatizing physics.

If you don't mean that sense, then what sense did you mean when you said "Physics has not been axiomatized at all"?

[,,,] That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. You pointed out repeatedly that what I said was not literally correct. I conceded that "plenty" of mathematicians disagree with what I said. Yet this is still not enough for you. — fishfry

I don't know what hyperbole you have in mind. Maybe 'nobody'. Because you seemed adamant with all-caps, and, as I recall, three variations of 'no', I didn't know it was hyperbole. So I merely replied to it at face value. Of course I would not have begrudged you then declaring it was only hyperbole. But still, I don't think what was hyperbolized was correct, even if given non-hyperbolized restatement.

Anyway, your response again misses my point. My point that you just quoted is not to take issue with your hyperbole, but rather to point out how your more recent argument goes wrong.

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Okay, but my point quoted above was not about that.

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it.
— TonesInDeepFreeze

I used a figure of speech called hyperbole. — fishfry

Do you mean the hyperboles "blow up the moon" and "AIDS denier"? If so, that's fine that you say now it was hyperbole. But I did take your comments at least to be a claim that a view that mathematical truth is not confined to model-theoretical is on its face preposterous even outlandish. I said that a lot of mathematicians don't view truth as merely model-theoretic, and you replied to the effect that there are intellectually talented people who believe a number of crazy things. It is reasonable for me to say that believing that truth is not merely model theoretic is not that kind of crazy, if it is even crazy at all.

I'm sure the standard axiomatization of math is an overkill for that. — fishfry

Some set theorists have pointed how we can reduce some axiomatic assumptions and still get the mathematics for the sciences. And even if ZFC is too productive, that doesn't refute that a good part of the interest in the axiom of infinity is to axiomatize (even if too productively) the mathematics for the sciences.

You don't know at all if it's been axiomatized ? — fishfry

No, not that I don't at all know. Rather, I don't know that it hasn't been axiomatized at all (i.e. hasn't been axiomatized to any extent whatsoever or with no progress toward axiomatizing it). That was in response to what you wrote, "Physics has not been axiomatized at all."

I'll go with "plenty" if this will mollify your sense of right and wrong here. — fishfry

I don't seek to be assuaged. You don't need to assent to 'plenty' on my account. Rather, one can assent to it merely on the grounds that it is obvious.

If you want to argue about what people think, I can't engage on that anymore. — fishfry

Yet you write:

99% of professional mathematicians are not involved in foundations (more or less objective number, I didn't look it up but recall jgill's post regarding the percentage of recently published papers) and therefore have no professional opinion on the subject at all. — fishfry

You continue to miss the point. That a vast number of mathematicians don't care about foundations doesn't imply that the vast number of mathematicians don't think axioms are true except model-theoretically, as indeed the fewer who care about foundations then reasonably we would expect the fewer who think truth is merely model-theoretic.

You made your point then got tedious and are now beyond that. — fishfry

It may have been tedious, but my entries have been responses to your continued posting on the subject. It seems odd to me that one would continue to reply, and repeat some points that are essentially the same, but then complain that responses on point to your own replies are thereby tediousness.

Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist.
— TonesInDeepFreeze

The question of foundations is as far from the practice of most mainstream mathematicians as blowing up the moon or AIDs denialism. — fishfry

Whether that is the case, my point is that having a foundational view that there is mathematical truth other than model-theoretic is not remotely outlandish in the class of advocating that we destroy the moon or that AIDS does not exist.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case".
— TonesInDeepFreeze

The former being interesting, the latter tedious beyond belief. — fishfry

You claimed the latter, so it is reasonable to reply to it whether you find that tedious or not.

So you didn't change the subject after all. — fishfry

I addressed the additional matter of outlandishness; I didn't thereby declare that I am forever changing any subject or not changing it, especially as you continued to post as if I had not already made clear that the question of what people think is distinct from whether they are wise to think it.

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.
— TonesInDeepFreeze

This could never be true. Physics has not been axiomatized at all. — fishfry

I didn't say "axiomatization of physics". I said "axiomatization of the MATHEMATICS for the sciences" [all-caps added]. Of course, though there is some consensus that set theory does axiomatize the branches of mathematics needed for the sciences, one may question whether indeed all of the needed mathematics is captured. But even a negative answer would not refute my point that among the salient reasons for adopting the axiom of infinity, at least we may say those reasons include an intent to lend support to axiomatizing the mathematics for the sciences, which is far beyond merely adding it to make things interesting. Also, I don't know that physics has not been axiomatized "AT ALL" [all caps added].

[the axiom of infinity] is just a rule that's been found by experience to make the game interesting. — fishfry

If 'to make interesting' includes 'to provide an axiomatization of the mathematics for sciences'.

if you deny that the axiom of infinity is "manifestly false about the real world," — fishfry

I neither denied it nor affirmed it.

Two different things: (1) "P is the case" and (2) "Nobody claims that ~P is the case".

Today when I read "Nobody thinks Colbert is funny", my first thought was not "But Colbert is funny" nor "I agree that Colbert is not funny", but rather how ludicrous it is to start an opinion article about American society with such a manifestly false claim as "Nobody thinks Colbert is funny."

The axioms aren't false, either, any more than the way the knight moves in chess may be said to be true or false. It's just a rule that's been found by experience to make the game interesting. — fishfry

That's your view. My point is not nor has been to convince you otherwise. Rather, my point is that no matter that it may be your view, it is not true that nobody (or only a few) people disagree with it.

I truly can't argue about what the majority or substantial plurality or "some" or "a few" or whatever mathematicians believe. — fishfry

Fine. And so there's not basis to claim that nobody (or merely a few) views axioms as true in a sense other than relative to models.

I have no data or evidence, neither do you. — fishfry

I have evidence from writings, conversations, and posts. From those, it is manifestly clear that it is false that "Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true."" Then, as to what the majority of mathematicians believe, I've stated my impression based on what I have read and heard from mathematicians, while I've said that of course that impression is not scientific.

But the subject matter that most mathematicians work on, as evidenced by the number of papers published, is so far removed from foundations that I can't imagine that many mathematicians spend five minutes thinking about the subject in a year or in a career. — fishfry

Again, that is the wrong road of argument for your position. I don't doubt that the vast majority of mathematicians don't care about foundations, in particular the model-theoretic notion of truth. But that only adds to my argument, not yours. Clearly, commonly mathematicians speak of the truth of mathematical statements, and even many mathematicians not occupied with foundations understand axioms in their field of study and often enough even the set theory axioms. So when such mathematicians say things like "the fundamental theorem of arithmetic" is true, then they don't mean it as "the fundamental theorem of arithmetic is true only in the sense that it is derivable in a consistent formal theory so that it is true in some models".

arguing popular opinion is not fruitful — fishfry

It's become a point of contention only because I responded to your claim about it, and not just in popular opinion, but your claim of totality of opinion.

Would you like me to go from "a few" to "a whole bunch?" I'm not sure what outcome would satisfy you. This is not a meaningful conversation. — fishfry

I don't care what you go to. I am making my own point that it is not the case that NOBODY (or even only a few) people regard axioms as true other than model-theoretically.

This is not a meaningful conversation. — fishfry

If it was meaningful for you to make the claim, then it is meaningful for me to reply to it, and to reply to your replies.

So 'real world' is now added to the question.
— TonesInDeepFreeze

That's what true and false typically mean. — fishfry

Views of mathematical truth don't have to be limited to what is typical otherwise. Whether or not departures from "typical" are justified, my main point was that it is not the case that all (or nearly all) mathematicians regard truth as merely model-theoretic.

Moreover, some mathematicians do regard certain mathematical statements in what is arguably a typical sense of finitary combinatory statements being concrete and true. And, as I mentioned, validities are true no matter what the models.

The axiom of infinity is manifestly false about the real world. — fishfry

That's your view. But it doesn't refute my point that it is not the case that all (or nearly all) mathematicians and philosophers regard axioms as true only as pertains to models.

Perhaps you can put your concept of truth into context for me. — fishfry

It doesn't matter toward my point. I have not claimed nor disagreed with any notion of truth. I don't have to just to point out that it is not the case that nobody regards axioms as true except relative to models. This reminds me of an article I read today. The writer claimed that nobody finds Colbert funny. I don't have to opine whether Colbert is funny to point out that it is false that nobody laughs at his jokes.

Respected mathematician Alexander Abian wanted to blow up the moon; and prolific author of high-level math texts Serge Lange was an AIDs denier. — fishfry

That opens another question. Whether one agrees with notions of mathematical truth other than model-theoretic, I'd be inclined not to claim that thinking philosophically or heuristically of mathematical truth as rather than model theoretic is among the ilk of proposing detonation of the moon or claiming that AIDS doesn't exist.

A progression of views (not necessarily your own):

(1) "Hilbert said that mathematics is only a meaningless game of manipulating symbols."

False. Hilbert was very much concerned with the contentual aspect of mathematics.

(2) "Al mathematicians view mathematics as only a meaningless game of manipulating symbols."

Clearly false.

(3) Formalism in mathematics is the view that mathematics is only a meaningless game of manipulating symbols.

False.

(4) There is a form of extreme formalism that views mathematics as only a meaningless game of manipulating symbols.

True.

(5) All mathematicians and philosophers hold that truth in mathematics pertains only to truth per models in mathematical logic.

Clearly false.

(6) All mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.

Clearly false.

(7) Most mathematicians and philosophers who understand truth per models hold that there are no viable senses of mathematical truth other than that of models.

Not known. My impression is that it is false, but would deserve a poll.

(8) Among mathematicians who know nothing, or very little, about models in mathematical logic, all (with possibly only few exceptions) regard axioms (incuding Peano, set theory) as true only with regard to models.

False, essentially a contradictory claim.

(9) Among mathematicians who know nothing, or very little, about models in mathematical logic, only a few are familiar with the set theory axioms.

Not clear. My guess is that it is false.

(10) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as false.

Almost surely false.

(11) Of those mathematicians who are familiar with the axioms of set theory, all (with possibly only a few exceptions) view the axioms as meaningful only as syntactic objects for syntactically proving other syntactic objects.

Clearly false.

/

In any case,

Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true." — fishfry

is false.

Which requires the existence of an inaccessible cardinal, the existence of which is not even provable in ZFC. — fishfry

Yes, which makes it even more curious what one would mean by saying the axioms of ZFC are false, while proposing a theory that is equivalent to ZFC PLUS another axiom.

tiny percentage of overall math papers that are devoted to set theory. — fishfry

That doesn't entail that a lot mathematicians aren't aware of axioms, including those not of set theory and those of set theory. And, again, probably most mathematicians don't get hung up on mathematical logic and its model theoretic sense of truth, yet mathematicians speak of the truth of mathematical statements.

And it's not even a given that only a few mathematicians who do understand models in mathematical logic hold that there are other senses of truth, including realism, instrumental, true-to-concept, et. al. Indeed, we know that there are mathematicians who well understand mathematical logic but still regard a sense of truth no restricted to that of "true in a model".

We're arguing over what other people think, we can't ever get to the bottom of that. — fishfry

You made a clam about it. We don't have a scientific polling, but we can see that there are many people who don't think that mathematical truth is confined only to the model-theoretic sense.

In what sense could the Peano axioms be true in the real world? — fishfry

So 'real world' is now added to the question.

Again, that you view certain notions about mathematics to be untenable doesn't entail that there are not plenty of people who don't share your view

The overwhelming majority of working mathematicians are not set theorists or involved in foundations. They pay no attention to set theory and would be hard-pressed to even name the axioms. — fishfry

I would expect that there is a wide range of interest in foundational axioms among mathematicians - from no interest to intense interest. But even among mathematicians with only little interest in foundations, there are those who regard certain axioms as true without having to quality by saying "only relative to models". For example, there are mathematicians who regard the Peano axioms as true, without saying "but only relative to certain models".

Moreover, as a special case, the logical axioms are true in every model, so those are axioms that would be said to be true without qualification as to models.

The question doesn't come up. — fishfry

The question whether mathematical truth is merely model-theoretic doesn't usually come up in studies outside foundations. But the question whether a given mathematical statement is true or false comes up all the time. Indeed, the fact that a great many mathematicians don't even care about foundations leads to them saying about certain axioms that they are true (while they don't qualify "but only in certain models"). The axioms adopted in the field of study are often regarded as true, even without bothering about model-theoretic truth. Indeed, for a good number of mathematicians, it is repugnant to view mathematics as just symbol study with the formulations not expressing mathematical truths unqualified by models.

Among those who study foundations, it must be abundantly clear that the axioms are arbitrary and not literally true — fishfry

What you say must be is just not the case. There are indeed mathematicians in foundations and philosophers who regard certain axioms as true without having to add "but only in certain models". This is not being ignorant of model-theoretic truth, but rather to view that there are other senses of truth - mathematical realism (Godel being the most famous), operational, et. al, or even just a naive sense in which mathematicians regard certain axioms and theorems to express truths in their fields of study, ranging from concrete finitary truths to greater abstractions.

or at best I would say that "a few" mathematicians claim the axioms are literally true in some sense. — fishfry

Then that is at lease some movement from 'NOBODY' [all caps original]. Of course for an accurate quantification, we would need an accurate poll. But from my readings in mathematics, from conversations with mathematicians, and from reading posts of mathematicians, I have found that there are many who regard certain axioms and theorems to be true without having to qualify to models. And my impression, though not derived by polling, is that that is the case for most mathematicians. Ask some mathematicians "Is the fundamental theorem of arithmetic true?" Then when the answer is 'yes', ask "Do you mean it's true only in certain models of the language of arithmetic, or do you mean it's just true"? I bet you dollars to donuts that most would answer 'just true' to the second question.

Also, adding "literally" narrows your first claim. There are notions of truth including, model-theoretic, literal, realism, operational, true-to-a-concept, et. al.

Among philosophers, who could seriously argue that the axioms of set theory are "true" in any meaningful sense; or even meaningful in any meaningful sense! — fishfry

You may think it not wise to think that they are true or meaningful in a sense other than model-theoretic, but that does not entail that no mathematicians or philosophers (or even only a few) have that view. Indeed, for example, mathematical realism, broadly put, is the view that there are mathematical objects and truth about them independent of consciousness about them. Ordinarily, mathematical realism would regard that there are true axioms that are true even without having to qualify "but only relative to certain models". For a good number of mathematicians and philosophers, they recognize that sentences that are not validities are not true in every model, but they still regard certain axioms to be true in a sense other merely model-theoretic..

I'd go further and say that it's perfectly clear that some of the axioms, such as the axiom of infinity, are literally false. — fishfry

I don't know what your definition of 'literal' is when it comes to mathematics, but you are of course entitled to your own view about certain axioms, but that does not entail that no (or even only few) mathematicians share your view.

I would add to all that the growing importance of neo-intuitionist, constructivist, and category-theoretic approaches, in which set theory is not only false, but irrelevant. — fishfry

Yet people do work in constructivist and intuitionist set theory. I don't know a lot about category theory, but it can be axiomatized by ZFC+Grothendieck-universe.

you said, "certain axioms," and I suppose if you want to say that high school notions like unions and intersections are true or instantiable in the real world, you'd have a point. — fishfry

I mean only that there may be axioms some mathematicians don't believe to be true. For example, some mathematicians might regard the axioms of Peano arithmetic to be true but not, the axiom of choice, or whatever. I don't mean to say that those axioms that mathematicians do believe to be true don't include set theory axioms such as power set, schema of replacement, infinity, choice, and even some who believe that the continuum hypothesis is true and some who believe the negation of the continuum hypothesis is true, or certain large cardinal axioms.

Your claim was that 'NOBODY' [all caps original] believes axioms are true without model-theoretic qualification. Now it is that at best only a few believe axioms are "literally" true. 'literal' is not defined yet in this conversation as to mathematical truth, and we should expect that different mathematicians would have different definitions if you forced them to respond to it, but it is not the case that only a few mathematicians and philosophers hold that there is no mathematical truth other than model theoretic, and, it is at least my impression that most mathematicians and a fair number of philosophers do hold that there is mathematical truth other than the model-theoretic sense.

Nobody claims that the axiom of replacement or the axiom of powersets is true. NOBODY [all caps in original] says that. [,,,] [no one] is claiming the axioms of math are "true." The axioms are strings of formal symbols, true in some models and false in others. — fishfry

Many mathematicians and philosophers of mathematics regard certain axioms and theorems to be true not just relative to models. It might even be the dominant view.

It might be fair to say that for Hilbert the syntax of logic does not include content. But Hilbert did not consider content irrelevant for mathematics. Indeed, Hilbert was very concerned with what he called the 'contentual' aspect of mathematics. Consistency and independence are syntactical, so Hilbert emphasized that they can be regarded irrespective of content, but mathematics also includes concerns other than consistency and independence, and Hilbert was keenly interested in the contentual, particularly finitary mathematics and application of mathematics to the sciences. Moreover, the relational structures themselves are an aspect of meaning. It is often said that Hilbert claimed that "mathematics is just a meaningless game of manipulating symbols". But I have not found any reference to a piece of writing or speech in which Hilbert said such a thing. In context of evaluating correctness of the application of the axioms and rules, Hilbert said that they may as well pertain to beer mugs as to anything else, but he did not say that therefore mathematics is devoid of content, as indeed he made clear that he thought quite the contrary.

/

In mathematical logic, of course, any set of formal sentences is an axiomatization of a formal theory. But many (maybe the preponderance of) mathematicians regard the axioms for particular fields of study not to be merely arbitrary, but rather as meaningful and true.

Thanks very much. I appreciate it.

Likewise your persistent complaint that I omitted the fact that I am talking about total orders (which you called "connected" for reasons I didn't understand). — fishfry

I didn't complain. I merely added the information.

And I used 'connective' in line with the notion of a connected relation.

I [...] decided to implicitly assume total orders to make the exposition more readable. — fishfry

I don't begrudge you striving for readability. I just wanted to add the point of clarification, which I I did by saying if x not equal y then either Rxy or Ryx. (The reason I used 'connective' rather than connected is that I wasn't clear in the moment whether 'connected' is an adjective applied to relations or to the set that the relation is on. No harm though, as I mentioned that 'connective' might be ersatz and so I defined it explitiy). If you wish to leave certain things as implicit in your own posting, then that should not preclude me from making them explicit.

If you would take a moment to ask yourself, "How would I explain ordinal numbers in a fair amount of depth to a casual audience," you might come to understand some of the tradeoffs involved. — fishfry

I think of tradeoffs virtually every time I post, since in such a cursory context of posting, I too have to take shortcuts. The fact that I added clarifications and information doesn't entail that I don't understand that an overview can't cover every technicality.

Necessarily not everything was perfectly pedantic. So you missed that point entirely. — fishfry

No, I got your point that your posts are meant only as an overview. But that doesn't entail that I can't mention clarifications and some more exact formulations myself.

A number of your statements were flat out wrong, — fishfry

If they are, then I'm happy to correct them.

such as claiming that a bijection of a well-ordered set to itself is necessarily another well-order. — fishfry

I specifically said the permutation is not a well-ordering. What is a well-ordering is the ordering induced. Again:

If R is a well ordering on S, and f is permutation of S, then R* is a well ordering on S as R* is defined by:

R*f(x)f(y) <-> Rxy

Clearly, by basic set theory, R* is a well ordering on S. (Indeed, <S R> is isomorphic with <S R*>.).

I have no idea why you would deny that.

I had already given the counterexample of the naturals and the integers. — fishfry

Whatever you said about the naturals and integers couldn't refute the theorem I just mentioned above.

Your several posts to me seemed not just pedantic, but petty, petulant, and often materially wrong. — fishfry

There is good reason for the various points I mentioned - they keep things clear, not merely pedantic. 'petulant' is psychologizing that happens to be incorrect. And I am happy to correct any errors I wrote.

You either misunderstood the pedagogy or the math itself — fishfry

I don't claim to be pedagogically expert. I posted to clarify certain points and to keep my mind focused a little bit on math occasionally.

a long list of topics to be studied before one can read my article. — fishfry

I said no such thing that your post couldn't or shouldn't be read without first studying anything. I said that a clear understanding requires understanding certain definitions. That does not preclude that one can first read your post then go on to learn the definitions. You are reading into what I wrote things that I did not write.

The challenge is to write something that can be read by casual readers WITHOUT any mathematical prerequisites. — fishfry

That's fine; I didn't write anything that begrudges you from doing that.

"Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem." followed by some picky complaint. — fishfry

It wasn't a complaint. I merely wished to add a clarification, as I said "just to be clear" that the theorem you mentioned doesn't happen to be the definition. That is relevant as someone might misconstrue that it was intended as a definition.

I led with "x is an ordinal iff x is the order-type of a well ordered set" because that's something that I can explain to a casual audience in a couple of paragraphs. — fishfry

That's fine, and I didn't fault you for it. I merely added a point of clarification.

With zero and 1, I take it a person can get to any number in {0, 1, 2,.., n}, though perhaps not efficiently. The limit of that being ω. — tim wood

Just to be clear, in set theory, the existence of a set that has all the natural numbers as members is not proven by taking a limit or a union.

Rather, from the axiom of infinity and axiom of separation it follows that there is a unique inductive set (a set that has 0 and is closed under the successor operation) that is a subset of any inductive set. That unique set is the set of natural numbers = w.

It is important to mention this so that it does not appear as if w is conjured in some non-rigorous way as "just gather together all the numbers you get starting from 0 and adding 1".

all the successors have already been used — tim wood

Within any limit ordinal, there is no last successor. For example, w is a limit ordinal, and there is no member of w that is the last successor.

we can both agree that mere familiarity with terms doesn't get a person very far. — tim wood

My point was not that understanding definitions is sufficient, rather that understanding definitions is necessary.

By "successor" I understand some number, as 3 is the successor to 2, 4 to 3, and so forth. — tim wood

'successor' for ordinals is simply this, by definition:

successor of x = x u {x}.

Defintions!

is the change from ω-street to ε-street a "can't get theah from heah" transition? I see the language that says you just add a successor, but what successor would that be? — tim wood

I think fishfry addressed that. epsilon_0 is a limit ordinal, not a successor ordinal. epsilon_0 is the union of the set of ordinals of the form w^x, where x is a finite sequence of ascending 'w' exponents. That is, epsilon_0 = U{w, w^w, (w^w)^w ...}.

ω is N in its usual order. — fishfry

w = N. No matter what order.

That does not contradict that also w is the order-type of <w standard-ordering-on_w> = the order-type of <N standard-ordering-on_N>

ω+1 is N in the funny order: — fishfry

That is plainly incorrect.

w+1 is not N, no matter what order.

w+1 = w u {w} = N u {N}

That does not contradict that w+1 is the order-type of <N "funny-order"-on_N>.

ω+1 as an alternate ordering of the natural numbers — fishfry

Again, w+1 is not an ordering. Rather w+1 is an ordinal, and it is the order-type of <w "funny-order"-on_N>.

The only way I can understand ω or ω+1, is simply as the numbers ω and ω+1, which are just larger than any of the {0, 1,.., n}. — tim wood

You are skipping the definitions:

w = the set of natural numbers

w+1 = w u {w}

Maybe I need a bit more care in thinking about what a number is. — tim wood

Mathematics doesn't have a separate definition of 'number' in general. Only definitions of 'number' with an adjective such as 'natural', 'rational', 'real', 'complex', 'ordinal', 'cardinal', et. al. So don't think of having to understand a concept of "is a number" onto itself.

When we say 'ordinal number', 'cardinal number', et. al, we could just as well merely say 'ordinal', 'cardinal', etc. The word 'number' is just artifact really. If 'number' is tripping you up, then forget about the word 'number'.

there are many who consider such books "archives" and that the sine qua non of learning is the good teacher. I myself favor middle ground, finding books to allow triangulation on a topic by one providing illumination where another is dark, while a teacher provides guidance and explains difficulties. — tim wood

Whatever many people may think, such books are key to understanding. But of course, a combination of books and teachers is best. In any case, in mathematics, to have a good understanding, it is crucial that the study is systematic - moving from the most basic concepts, step by step through definitions and proofs, to more complicated material.

I vaguely remember a comment that "explained" certain large cardinals by saying, "You can't get theah from heah." By which I understood that no ordinary process could get to them, meaning, as best I get it, that no recursion scheme could get to them. — tim wood

I take it you mean 'large cardinal' in the mathematical sense. I don't know enough about the relation of recursion and large cardinals, but I would be careful saying things like "no recursion scheme could get to them" without specifying exactly what you mean by that. In any case, the context at least requires understanding the definitions of 'large cardinal' and 'transfinite recursion'.

Every permutation of a finite set is a well-order. — fishfry

For clarity, I prefer to use 'permutation' in its exact mathematical meaning. A permutation is a bijection from a set onto itself. In that regard, a permutation is not an ordering. A permutation may "induce" (I don't know whether there's a more usual terminology) an ordering:

Any permutation f of a set (finite or infinite) that has a well ordering R induces a well-ordering R* as follows:

R*f(x)f(y) <-> Rxy

A well-order is an order in which every nonempty subset has a smallest element. — fishfry

Again, we need also to include mention that the order is a connective order.

So for infinite sets, permuting does not preserve order type. — fishfry

With the ordinary mathematical definition of 'permutation', the above is incorrect. As mentioned above:

Any permutation f of a set (finite or infinite) that has a well ordering R induces a well-ordering R* as follows:

R*f(x)f(y) <-> Rxy

in the limit, or sup operation. — fishfry

We should mention that limits (aka 'sups') in regard to ordinals are unions that are not successors. We need to have a good understanding of both binary union and generalized union to understand ordinals.

Infinity and the Mind by Rudy Rucker. — fishfry

That is an entertaining book, but one might need to take it with a grain of salt regarding certain technical matters (I don't recall the particular matters now).

Another entertaining book about logic is William Poundstone's 'Labyrinths Of Reason'.

An ordinal number is the order-type of a well-ordered set — fishfry

Yes, 'x is an ordinal iff x is the order-type of a well ordered set' is a theorem. From the definition of 'order-type', every order-type is an ordinal. And with a trivial proof, every ordinal is an order-type. But, just to be clear, 'x is an ordinal iff x is the order-type of a well ordered set' is not a definition of 'is an ordinal', lest we have the circularity of defining 'order-type' with 'ordinal' and 'ordinal' with 'ordinal type'.

A well-order is an order relation on a set such that every nonempty subset has a smallest element. — fishfry

Not just any order relation such that every nonempty subset has a minimal member. Rather, a connective order such that every nonempty subset has a minimal member. ('connective order' might not be a usual terminology; what I mean is an order R on S such that for every x and y in S, either Rxy or Ryx.)

Every possble nonempty set of the naturals has a smallest member, so the naturals are well-ordered by <. — fishfry

Yes, but again, also because < is a connective order. Also, merely as a small note, we don't need to mention modality with 'possible'.

the negative numbers have no smallest element. — fishfry

Yes, but just to be clear (since we are considering contexts in which we mention also other orderings), the set of integers has no minimal element regarding the standard ordering on the integers.

ω stands for the natural numbers in their usual order. — fishfry

w is the order-type of <w standard-ordering-on_w>, but more basically, w is the set of natural numbers, even irrespective ordering.

If the usual order 0, 1, 2, 3, ... is called ω — fishfry

w is not an order. w is the order-type of <w standard-ordering-on_w>.

our funny order (N,≺) is called ω+1 — fishfry

w+1 is not an order. w+1 is the order-type of <w ≺>.

even without choice, some uncountable sets can be well-ordered. Just not all of them. — fishfry

It is not the case that the absence of choice implies that not all sets have a well-ordering. Rather, the absence of choice leaves it undecided whether all sets have a well-ordering.

in the absence of choice, there are infinite sets that are not well-ordered — fishfry

As above.

there's also no set of all sets bijectively equivalent to the naturals or any other cardinality. — fishfry

Except the cardinality 0. The set of all sets that have a bijection with 0 is {0}.

the Alephs are indexed by whole numbers — fishfry

As you mentioned later, for any ordinal, there is the aleph indexed by that ordinal.

I didn't say one needs to be a specialist. Having an adequate grasp of the basic terminology doesn't require that one be a specialist. And one can get the "general idea" about ordinals, but a clear understanding includes a clear understanding the definitions of the terminology used.

Here is some of the terminology (not necessarily in logical order) that one must have a very clear understanding of in order to have a clear understanding the matters in this thread. If one is at all unclear about this terminology and the axioms and main theorems, then confusion is virtually inevitable. But a good textbook takes you through it all step by step:

class
set
empty set
proper class
subset
power set
union
pair
ordered pair
singleton
relation
domain
range
field
Cartesian product
converse relation
function
bijection
into
onto
permutation
reflexive relation
irreflexive relation
symmetric relation
anti-symmetric relation
asymmetric relation
transitive relation
connected
linear ordering
well ordering
epsilon transitive
ordinal
successor ordinal
limit ordinal
transfinite recursion
homomorphism
isomorphism
order-type
finite
natural number
set of natural numbers
infinite
Dedekind infinite
countable
uncountable
cardinal
cardinal addition
cardinal multiplication
cardinal exponentiation
aleph
ordinal addition
ordinal multiplication
ordinal exponentiation
recursive ordinal
epsilon_0
tuple
lexicographic ordering

Is it correct to think of all the well-orderings to be the same thing as all the permutations? — tim wood

No. The set of all permutations of S is the set of all bijections from S onto S. The set of all well orderings of S is something different.

ω! — tim wood

For a natural number n, we have n! = the cardinality of the set of all permutations of n. But n! is not the set of all permutations of n.

So I would take 'w!' to mean the cardinality of the set of all permutations of w.

NN — tim wood

By 'NN' I surmise you mean the set of natural numbers. That's confusing notation. I would just use 'N' or 'w' [read as omega].

seems that ω is the ordinal associated with NN. But it also seems that ω is also associated with every infinite subset of NN. — tim wood

w = N = card(N) = aleph_0 = the order type of <N standard-ordering-of_N>

'w', 'N', 'card(N)', 'aleph_0' are all names for the same set.

If S is an infinite subset of N, then w = card(S)

how many infinite subsets of NN are there? — tim wood

First, the cardinality of the set of infinite subsets of N is the cardinality of the power set of N (because the set of all finite subsets of N is countable, and the cardinality of the union of a countable set and an uncountable set is the cardinality of the uncountable set).

So the question is:

What is the value of x in the following?:

card(power set of S) = card({f | f is a function from S into {0 1}) = aleph_x

The answer to that depends on the continuum hypothesis.

ε — tim wood

What does epsilon stand for there? Do you mean epsilon_0?

how do you get beyond ε and still be countable? — tim wood

epsilon_0 + 1 is countable. There is no greatest countable ordinal.

/

To keep in mind:

There are two different things: (1) a set S and (2) <S R> where R is a well ordering of S. Strictly speaking, it doesn't make sense to say 'S is not well ordered'. Rather, strictly speaking, we would say for a given R that is not a well ordering of S that "S is not well ordered by R'. For example, strictly speaking, we would not say "The integers are not well ordered" but instead, strictly speaking, we would say "The integers are not well ordered by the standard ordering of the integers". In casual contexts, we take it for granted that we are referring to the standard orderings, so "the integers are not well ordered" is okay in that casual context. But sometimes we need to be stricter to avoid misunderstanding.

Also, strictly speaking, permutations themselves are not usually well orderings, but rather a permutation may "induce" a well ordering. A permutation is a bijection from a set onto itself. That is not usually a well ordering. In casual contexts, we benignly conflate the permutation with an ordering the permutation "induces", but sometmeise we need to be stricter to avoid misunderstanding.

/

General suggestion: It is virtually impossible to gain a clear understanding of set theory through back and forth posts. Only a textbook studied systematically starting at page 1 is likely to work.

First? Or least but not necessarily first? — tim wood

'first' in the sense that there is no member that precedes it in the ordering. Usually we say 'least' or 'minimal'.

can be and is — tim wood

In ordinary mathematics, other than for colloquial emphasis, there is no difference in meaning between 'can be' and 'is'.

well-ordering means the set [...] is ordered lexicographically. — tim wood

The definition of 'R is a well ordering of S' is:

All three hold: (1) R is a subset of the set of ordered pairs of members of S, and (2) for any different members x and y of S, either Rxy or Ryx, and (3) for any non-empty subset of S there is a member m of the subset such that there is no member x of the subset such that Rxm.

The definition of 'There is a well ordering of S' (or 'S is well ordered') is:

There exists an R such R is a well ordering of S.

Lexicographic ordering is something different.

1,2,3 is well ordered in each, and all, of six variations — tim wood

Every permutation of {1 2 3} "induces" a distinct well ordering of {1 2 3}.

A permutation of a set is a bijection from the set onto itself. For each permutation f of {1 2 3} there is the well ordering R on {1 2 3} defined by;

Rxy <-> (x e {1 2 3} & y e {1 2 3} & f(x) < f(y))

Further, in greater generality, a well ordering R of a set S is relation such that both (1) R is a subset of the set of ordered pairs of members of S, and (2) for any different members x and y of S, either Rxy or Ryx, and (3) for any non-empty subset of S there is a member m of the subset such that there is no member x of the subset such that Rxm.

Then we see that S is an ordinal if and only if both (1) every member of a member of S is a member of S, and (2) the membership relation on S is a well ordering of S.

Then, along the lines of fishfry's post, it is a theorem of ZF (transifinite induction is used) that for any set S and well ordering R of S, there is a unique ordinal D, such that <S R> is isomorphic with <D membership_relation_on_D>. So D is called 'the order type of <S R>'.

An ordering has a first element (yes?) — tim wood

A well ordering of set S provides that every non-empty subset of S has a first element. And S is a subset of S, so if S is non-empty, then S has a first element. The first element of any ordinal is 0 (the empty set).

But there are orderings other than well orderings.

At some point candidates for the second element are exhausted, — tim wood

Not if the set is infinite.

Also, if this is pertinent to your question, bear in mind that some ordinals have no last element and some ordinals do have a last element.

uniquely orderable — tim wood

What does "uniquely orderable" mean?

Re:

The question was asked by tim wood: "What is an infinite ordinal?"

As direct an answer I can provide:

S is an ordinal if and only if all three: (1) every member of a member S of is a member of S, and (2) for any different members x and y of S, either x is a member of y or y is a member of x, and (3) in every non-empty subset of S there is a member m in the subset such that no member of the subset is a member of m.

S is infinite if and only if there is no 1-1 correspondence between S and a natural number.

S is an infinite ordinal if and only if both (1) S is an ordinal, and (2) S is infinite.