3

Books on set theory seem to at least strongly imply that there can exist some $\mathsf{ZFC}$ universe whose $\omega$ is "standard" in an "absolute" sense (order-isomorphic to the concept of "standard $\mathbb{N}$" described, for example, in the lecture notes here and here). However, it's not totally obvious to me why this is necessarily true. Granted, if one accepts a multiverse philosophy of set theory, then presumably Lowenheim-Skolem guarantees the existence of universe(s) which are countable (when viewed "from the outside"), and the $\mathsf{ZFC}$ axioms guarantee that such universes each have some smallest inductive set (the set $\omega$). But this by itself doesn't seem to preclude the possibility that these all have an $\omega$ that's order-isomorphic to a non-standard (countable) model of Peano arithmetic. Timothy Chow's website features this excerpt from a (somewhat aged) newsgroup thread touching on this issue:

Consistency doesn't buy you everything that you might want from a formal system---it tells you that models of the system exist, but it doesn't tell you that there are any models where the "integers" are isomorphic to the standard integers, so you can't necessarily "trust" theorems of the system

So my question is: What exactly is the rationale for having some level of confidence (belief?) that there actually can exist a $\mathsf{ZFC}$ universe whose $\omega$ is "standard" in the aforementioned "absolute" sense? (i.e. that such a universe can actually satisfy all the axioms)

  • The difficulties around actually formalizing this notion of standardness of $\omega$ for any particular universe (see here) ostensibly poses some challenges here, but I would think that at least some informally reasoned rationale should be possible?

Footnote: Perhaps(?) some might argue that the question doesn't really matter, that we can just go ahead proving theorems while being agnostic about which kind of universe (standard or non-standard $\omega$) we're actually working in. But such an argument (if indeed anyone argues it?) seems like it would be inadequate for applied purposes, for example physicists needing to mathematically describe physical space. What is a square whose sides are a non-standard number of meters in length? What is a polygon with a non-standard number of sides? Such notions seem nonsensical for any conceptual model of physical space that I'm aware of.

Z. A. K.
  • 11,359
NikS
  • 869
  • Every non-standard Peano model contains the standard one. And therefore cannot be $\omega$, defined as the smallest one. In fact order type of countable non-standard models is known. Besides these are not even well ordered to begin with (only linearly). – freakish Jan 11 '24 at 08:34
  • 1
    Having followed your series of questions, I doubt this admits an acceptable answer. The notion of "absolute standard $\mathbb{N}$" in the sense of the numbers we all learned to count with in elementary school is incoherent / a category error. I have yet to see evidence that standard model is ever used to mean anything that is not covered by "the set $\mathbb{N}$ with its usual addition and multiplication in the set theory you use as your metatheory", and whether that corresponds to the numbers you use in primary school is not a mathematical question, but a philosophical/scientific matter. – Z. A. K. Jan 11 '24 at 08:52
  • @Z.A.K. : Sounds like (if I understand correctly?) your argument is: the $\omega$ of the universe I’m working in might “look like” the numbers we learned to count with in school, or it might be some more complicated linear order, but asking “to which of these possibilities does my universe’s $\omega$ actually correspond?” isn’t even a legitimate mathematical question. I guess I would be a bit surprised if that were a consensus view, but I can’t claim to really know. The excerpt I quoted from Tim Chow’s website would seem to imply a contrary view. – NikS Jan 11 '24 at 10:42
  • @NikS, see Tim's response at the first page I linked in my answer :-) – Mikhail Katz Jan 11 '24 at 11:05
  • @NikS: I've read Chow's linked text right before posting my comment to make sure that it does not make any claim about "standard integers" that would contradict the interpretation as "the set $N$ with its usual addition and multiplication in the set theory you use as your metatheory". And yes, everything works there under the "standard = metatheoretic $N$" interpretation: if your metatheory proves ZFC $\omega$-inconsistent, then $\exists x\in \mathbb{N}. P(x)$ can be a theorem of ZFC even if there is no ordinary integer $n \in N$ in the set theory you use as your metatheory satisfying $P(n)$. – Z. A. K. Jan 11 '24 at 11:59
  • @NikS: I think you're reading way more into the word "standard" than what it entails in ordinary discussions in logic and set theory. To the extent that standard integers are discussed in a mathematical context, they refer to the set of integers in the metatheory you're working in. To the extent that some strands of philosophy (e.g. object realism) posit a correspondence between physical phenomena and abstract math, one might speak about whether your metatheory's integers have this desired correspondence or not; that's no longer a question of mathematics, but physics (or philosophy). 1/ – Z. A. K. Jan 11 '24 at 12:13
  • @NikS: But in the latter case, using the usual model-theoretic setup and terminology would be misleading at best: when a model theorist talks about model, it means are abstract mathematical object in the metatheory that satisfies certain properties (validating the interpretation of the axioms of some logical theory). They're not talking about physical objects like square plots of land or number of LEGO cubes one might count in elementary school. There's solid consensus around that: I doubt you'll find any logician for whom model refers to anything like the latter. 2/ – Z. A. K. Jan 11 '24 at 12:17
  • @NikS: "Which side lengths can a square plot of land have?" is primarily a physics and philosophy question: you'll have to either empirically look and see, or make some assumptions about the physical world and how it corresponds to the mathematical world. In either case, it'll be a question for physicists and philosophers, and the answer need not look like a "model of the integers" at all: e.g. $10^{60} \in \mathbb{N}$ but what does it mean to have a square plot of land whose sides are $10^{60}$m long? I'm not sure, and I don't see how set theory (or logic) could help there :( 3/3 – Z. A. K. Jan 11 '24 at 12:24
  • @Z.A.K. : Maybe the physics examples aren’t the best, as they mix in non-mathematical issues about the nature of physical reality. A more purely mathematical question that gets at similar concerns might be this: Can there exist a Turing machine that ever accesses data cells on the “tape” which are separated from the starting position by a non-standard number of intervening cells? (seems like the answer should be “no”, though at this point I’m legitimately not 100% sure) – NikS Jan 12 '24 at 11:55

1 Answers1

1

"there can exist some ZFC universe whose ω is "standard" in an "absolute" sense": This is the standard model/intended interpretation hypothesis, that has been discussed in a number of posts; see e.g., this or that. The relevant distinction here is between the metalanguage integers, on the one hand, and the formal (object) language integers. The former can be thought of as a sorites-like subcollection (not subset!) of the latter. Attempts to identify them lead to much confusion.

Mikhail Katz
  • 42,112
  • 3
  • 66
  • 131
  • In order to define what first-order logic even is, we have to start with some (informal or semi-formal) concept of natural number (e.g. here) -- I assume this is what you mean by "metalanguage integers". Then, can we not say that the "metalanguage numbers" *are* one possible model of $\mathsf{PA}$ (or more precisely, that there exists a model of $\mathsf{PA}$ that's order-isomorphic to the "metalanguage integers")?...(cont'd) – NikS Jan 15 '24 at 09:08
  • ...Indeed, Carl Mummert's answer here (from your link) seems to describe exactly this when he speaks of the "'standard model' $\mathbb{N}$...from the perspective of a certain kind of realism." And so does Kleene's book (quoted in the answer) when Kleene describes the motivating "portions of existing informal or semiformal mathematics" as defining an "(intended or usual or standard) interpretation or interpretations of the formal system" and says that "the informal mathematics that we aim to formalize in $\mathbb{N}$ is elementary number theory". – NikS Jan 15 '24 at 09:08
  • @NikS, why do you say that "the 'metalanguage numbers' are one possible model of PA" ? They certainly cannot be said to do that. For one thing, they cannot be a model of PA (relative to the metatheory ZFC) simply because they don't form a set. – Mikhail Katz Jan 15 '24 at 10:14
  • "the informal mathematics that we aim to formalize in N is elementary number theory" : this is certainly true. However, this does not mean that N does not contain ideal elements that do not correspond to any naive integer; see https://mathoverflow.net/questions/231599/whats-reebs-take-on-naive-integers – Mikhail Katz Jan 15 '24 at 10:33
  • I guess another way to put it would be “there exists a model of $\mathsf{PA}$ for which every element of $\mathbb{N}$ can be expressed as some term in first-order logic whose last symbol is $\mathbf{0}$ and every preceding symbol is the successor function symbol $\mathbf{S}$” (i.e. $\mathbf{0, S0, SS0,}$ etc.) – NikS Jan 15 '24 at 10:43
  • How many $\mathbf{S}$'s are allowed? @NikS – Mikhail Katz Jan 15 '24 at 15:23
  • Well, I'm presuming there's no restriction on length, in the same way that the $\mathsf{ZFC}$ Axiom Schema of Separation says "for every formula $\phi(x)$" without presuming any restriction on the length of the formula $\phi(x)$ – NikS Jan 15 '24 at 21:06
  • Would you describe the length of such formulas as a metalanguage integer or an object-language integer? @NikS – Mikhail Katz Jan 16 '24 at 10:28
  • Metalanguage integer (per what I think is meant by the the term “metalanguage integer”) – NikS Jan 16 '24 at 12:34
  • OK then your $\mathbf{SS}$ merely rephrase the same problem we had before, namely that the metalanguage integers are a sorites-like collection and not a set, they cannot be said to form a model of PA. It is not easy to escape the basic dilemma, namely that we don't seem to have an absolute notion of what "finite" means; see e.g., this answer and the comments below it. – Mikhail Katz Jan 16 '24 at 12:47
  • This argument seems more or less equivalent to that of Georges Reeb that "the naive integers don't fill up $\mathbb{N}$. I took a look at pertinent sections of Diener & Reeb's Analyse Non Standard, your related paper, and this other one by Reeb (my French is not fluent, but "good enough" for the task, I think). However, I'm still not 100% sure what Reeb's argument is here. – NikS Feb 04 '24 at 08:57
  • Would I be correct in concluding that his conception of "naive integer" is subject to a kind of "limitation of size"? That seems to be the argument, i.e. that a "naive integer" can't be arbitrarily large because eventually you reach some sort of "limiting" order of magnitude, beyond which the concept of "naive integer" is no longer meaningful (due to, perhaps, physical considerations like the impossibility of building a computer that counts all the particles in the known universe). – NikS Feb 04 '24 at 08:58
  • @NikS, It is hard to know what he meant exactly because he published very little. As far as the "computer that counts all the particles etc." is concerned, I elaborated in this direction in my answer linked above. – Mikhail Katz Feb 04 '24 at 09:56
  • I see, your argument here related to a computer the size of the universe suggests a conception of the “naive integers” being indeed subject to the sort of “limitation of size” I mentioned in my just prior comment. Now if furthermore (per your comment below this) the “naive integers” are the number-concept of our metalanguage, which we use to speak of “lengths” of logic formulas, then presumably we should regard logic formulas as subject to this same “limitation of size.” Am I reading this correctly? – NikS Feb 09 '24 at 08:22
  • @NikS, Well most proofs do not exceed in length the size of the physical universe, so we are probably safe here. – Mikhail Katz Feb 09 '24 at 11:03
  • Certainly any proof explicitly written down (on paper or computer file system) is inherently limited by physical constraints. But the situation seems a bit different for, say, the Axiom Schema of Separation: said schema is conceptually comprised of a multitude of individual axioms (sentences), but there’s no presumption (I don’t think?) that each of them individually be physically written down…(cont’d) – NikS Feb 10 '24 at 04:08
  • …So the stipulation that the metalanguage numbers (“naive integers “) be subject to “limitation of size” seems to carry with it a definite philosophical implication for what $\mathsf{ZFC}$ *is, namely that the Axiom Schema of Separation defines a finite* multitude of individual axioms (“finite” in the sense of the naive integers’ “size limitation”). Correct? – NikS Feb 10 '24 at 04:15
  • @NikS, We can't really pin down exactly what "finite in the sense of naive integers" means exactly. As I mentioned (or should have mentioned), the collection (not set!) of naive integers forms a sorites-like collection that on the one hand seems to satisfy the inductive axiom (if $n$ is naive then apparently also $n+1$ should be naive), yet it does not form a a set. Axiomatic nonstandard analysis where one works within $\mathbb N$ itself (rather then extending it to the hypernaturals) can be thought of as a formalisation of the sorites paradox: the standard integers in $\mathbb N$ aren't a set – Mikhail Katz Feb 11 '24 at 09:53
  • It seems like there are two mutually exclusive concepts in play here: 1) Naive integers being subject to some “limitation of size” (some “limiting” order of magnitude, e.g. due to physical constraints like “computer the size of the universe”). 2) Induction on naive integers, i.e. for every naive $n$ there’s a naive $n + 1$. These two concepts can’t possibly *both* pertain to the naive integers, can they? – NikS Feb 12 '24 at 05:50
  • I think the discussion of limitation on size such as computer memory is merely a way of illustrating the problem with metalanguage integers which according to most accounts cannot be so easily identified with the object-language integers. If the naive integers are a sorites-like collection which is not a set, obviously you are not going to be able to pin them down the way you are trying to do. In the comment above, I did not mean to say that the standard integers inside Nelson's $\mathbb N$ (in IST) are the naive integers. It is merely one way of modeling them to capture the distinction. – Mikhail Katz Feb 12 '24 at 10:19
  • Regarding “not going to be able to pin them down”: That sounds like an assertion that the concept of “naive integer” has a vagueness around it that makes it not possible to say with definite certainty whether they’re subject to some “limitation of size”, nor possible to say with definite certainty whether induction is valid for naive integers (“$n$ is a naive int.” $\Rightarrow$ “$n + 1$ is a naive int.”). Am I reading that right? – NikS Feb 12 '24 at 22:34
  • 1
    @Nik, in the paradox of the sorites, it is taken for granted that if $n$ grains of sand do not form a "heap" then $n+1$ don't form one, either. So I would say that, similarly, the naive integers are assumed to satisfy induction, but one would nonetheless say that we don't know exactly what they are. They are of course modeled by the standard integers within $\mathbb N$ in Nelson's framework, and intuitively perceived in terms of "limitations of size" that you have mentioned. If you are trying to pin down a paradox in what I am saying, you've already succeeded: it's the paradox of the sorites! – Mikhail Katz Feb 13 '24 at 10:45
  • Ah, OK, I think I get the idea now . Thanks. – NikS Feb 13 '24 at 11:24