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This thread did confirm for me the concept of $\omega$-models vs non-$\omega$-models, the latter type of model having a smallest inductive set that is NOT order-isomorphic to the familiar "standard" natural numbers (but instead to some more complicated non-standard model of Peano arithmetic)

But this brings up a related question regarding the universe (the "universe $V$ of all sets", that the models are presumed to be subclasses of): Even discussions that explicitly acknowledge the possibility of models $\mathfrak{M}$ with non-standard $\omega^{\mathfrak{M}}$ seem to assume that the universe's smallest inductive set is guaranteed to be the standard $\omega$ (order-isomorphic to the "standard model $\mathbb{N}$" of Peano arithmetic). Is this really a valid assumption?

  • As a counter-example, imagine I have some model of ZF(C), $\mathfrak{M} = (\textbf{M}, \textbf{E})$, whose smallest inductive set $\omega^{\mathfrak{M}}$ is non-standard. Then it seems legitimate (in the same way you can say $V = L$) to say $V = \mathfrak{M}$. That is, conceptually, I throw away the sets that are not members of $\textbf{M}$, and then redefine the "true" $\in$ relation of the universe to match $\textbf{E}$. By construction, this is a set universe that satisfies the axioms, but whose $\omega$ is non-standard.

Footnote: An example of a book which (although it doesn't discuss models) does seem to assume that $\omega$ is always the standard $\omega$ is Halmos' popular Naive Set Theory (despite the title, it is, as explained in the preface, "axiomatic set theory from the naive [or informal] point of view"). This excerpt shows the discussion of natural numbers and $\omega$.

  • Elements of Set Theory by Enderton and Introduction to Set Theory by Hrbacek & Jech also seem to make the same assumption of $\omega$ always standard.
NikS
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  • Maybe you could bring up some example of a discussion like you mentioned in the second paragraph? My very naive idea is that one always assumes the base universe (or working universe) to be standard, because that's what we intuitively want, and we cannot meaningfully talk about it anyway. Inside the base universe, we can construct standard and non-standard models to examine them. – fweth Mar 27 '23 at 10:37
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    Well, the three books I mentioned in the “footnote” all seem to assume that $\omega$ is always the standard $\omega$. It could be (as you suggest) that unless explicitly stated otherwise, the Axiom of Infinity is assumed by default to imply the existence of “standard $\omega$”. Could be the case, but I’ve never seen that explicitly stated to be the default assumption. – NikS Mar 27 '23 at 10:55
  • I think statements like $V=X$ are generally more of a metaphysical exercise, like some people believe that there exists an actual $V$ and we can discuss its properties, while others say all we can do is derive from ZFC and see what we get. I think the first group, i.e. the platonists, in general believe that $V$ is nice, and that only first order logic lacks the power to actually describe all of its properties, like we cannot describe $\mathbf{N}$ so well that we can rule out non-standard models without using second order logic. But I'm not a logician so take everything with a grain of salt – fweth Mar 27 '23 at 13:00

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As it happens, this very point was the subject of a discussion John Baez and I had here. (Part of a long discussion on the $\omega$'s of models of ZF.) John's viewpoint is that "standardness" is a relative concept.

This question cannot be totally disentangled from philosophical issues (but there are some purely mathematical aspects, which I'll address below). If you're a strong Platonist (like Gödel), you believe there is a unique mathematical universe of sets "out there". Call it $V$; then by definition the standard $\omega$ is the $\omega$ of this $V$.

If you prefer to hedge your bets, but still take a more-or-less realist stand, then you might like what we called the "three-decker sandwich". There is some universe $V$, the "outermost" universe; standardness is relative to this. We make no ontological claims about $V$; maybe there are a variety of "universes" that could serve for $V$. Inside $V$ is a model $U$ of ZF, and $\omega^U$ is standard if it's isomorphic to $\omega^V$. The locution "the standard $\omega$" is rejected as unnecessary, and burdened with metaphysical assumptions.

Of course, if everything is developed relative to $V$, then it fades into the background. It's needed mainly to give meaning to the term "standard". (Also in locutions like "the standard $\in$".) The "three-decker sandwich" is $\omega^U\subset U\subseteq V$.

The three-decker sandwich helps explain how non-standard $\omega$'s "work". Consider $\omega^U$. We have $\omega^V\subset\omega^U$, indeed it's a proper initial segment. At the bottom of p.44, Halmos defines $\omega$ as the intersection of all the "successor sets" (aka inductive sets) contained in a set $A$. Now, $\omega^V$ is a successor set, but it does not belong to $U$. That's why $\omega^V$ is smaller than $\omega^U$, even though $V$ is larger than $U$. Informally, the "inhabitants" of $U$ cannot "see" $\omega^V$, and so they compute the intersection incorrectly.

What about the idea that $\omega$ consists of the elements that can be obtained by starting with $\varnothing$ and applying the successor function $n$ times, with $n\in$ "the standard model $\mathbb{N}$" of the Peano Axioms with 2nd-order induction? Halmos does not have a formal theorem expressing this. Rather, he says that $\omega$ provides a rigorous counterpart of the intuitive description that the natural numbers are 0,1,2,3, "and so on". Then in the next chapter, he shows that $\omega$ satisfies the 2nd-order Peano axioms. In a set-theory context, $\mathbb{N}$ is essentially defined to be $\omega$. The two notations emphasize different aspects of the set of natural numbers: $\omega$ as an ordinal, $\mathbb{N}$ as a free-standing structure, whose signature (in the sense of mathematical logic) would include + and $\cdot$ but not $\in$.

Back to philosophy. If you're some breed of formalist, then the universe of all sets is a fiction. All that matters is what you can prove in some formal system like ZF. In ZF, one can give a formula $\varphi(x)$ and prove that there is a unique set $w$ satisfying $\varphi(w)$; moreover, $\varphi(x)$ is a formalization of the usual definition for $\omega$. Statements mentioning "the standard $\omega$" are rephrased using $\varphi$.

No doubt there are many other philosophical positions one might take. Perhaps one accepts the "multiverse of universes" (like Joel David Hamkins), but feels all the “legitimate” ones have the same $\omega$. (This seems to be the view of Scott Aaronson.) The argument for this: our intuitions about the natural numbers seem much more concrete and solid than about the entire universe of all sets, which disappears into the misty heights with the higher cardinals. (And is fuzzy around the edges, thanks to things like the Continuum Problem.) Others might feel that there's plenty we don't know about $\mathbb{N}$; maybe the notion of "the standard natural numbers" is a bit foggy after all.

As for the textbooks, these usually keep philosophy to a mininum. A couple of quotes from Halmos' preface:

The purpose of the book is to tell the beginning students of advanced mathematics the basic set-theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism.

A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary...

I don't know what Halmos' personal philosophy was, but these quotes suggest a stylistic Platonism. As long as we're working in some fixed "outermost" set-theory universe $V$, we're entitled to use the notion of "the standard $\omega$". Pedagogically this is fine: it seems to have a clear intuitive meaning, and if we're not discussing non-standard models, might as well let sleeping dogs lie.

An analogy. Quantum mechanics boasts a large body of controversy over its proper interpretation. David Mermin once noted that most physicists in practice go by the "shut up and calculate" interpretation. With regard to these set-theory matters, I dare say most mathematicians are happy with "shut up and prove theorems" most of the time.

Michael Weiss
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  • Does that mean that ZFC can't prove that the smallest inductive set is order-isomorphic to $\mathbb N$? Or is that not even a statement the language of ZFC can make? – eyeballfrog Mar 27 '23 at 15:08
  • @eyeballfrog Usually, in the context of ZF, $\mathbb{N}$ is defined to be $\omega$. There are other possible formal interpretations of the statement "$\mathbb{N}$ is order-isomorphic to $\omega$", but ZF can prove them. – Michael Weiss Mar 27 '23 at 15:34
  • In the context of the "three-decker sandwich", according to $V$, $\omega^U$ is not minimal inductive. The initial segment isomorphic to $\omega^V$ is smaller and inductive. But $U$ "thinks" $\omega^U$ is minimal inductive because that initial segment does not belong to $U$. – Michael Weiss Mar 27 '23 at 15:39
  • That reminds me of the situation with countable models of ZFC, where the model can't "see" the bijection between $\aleph_0$ and $2^{\aleph_0}$. Similar idea? – eyeballfrog Mar 27 '23 at 15:42
  • Yes, exactly, it's the same idea. – Michael Weiss Mar 27 '23 at 16:29
  • Thanks @MichaelWeiss Relative to the "philosophical menu" that you postulated, I'm probably in the middle category ("more-or-less realist"). However, most(?) textbook treatments (e.g. Halmos, linked to in the question) make a fairly specific claim about the $\omega$ of $V$, namely: every element of $\omega$ can be obtained by starting with $\emptyset$ and applying the successor function $n$ times (where successor function is $S(x) := x \cup {x}$ and $n \in$ "the standard model $\mathbb{N}$" of the Peano Axioms with 2nd-order induction)...(cont'd) – NikS Mar 29 '23 at 07:09
  • ...So from the "more-or-less realist" perspective, presumably one would say that a maximally general set theory is obtained by considering a wide variety of possibilities for the nature of $V$ (a "multiverse" philosophy), but the introductory textbook treatment focuses on the "base case(s)" (simplest, most intuitive case(s)) in which $V$ has an $\omega$ conforming to the "specific claim" I just described. Am I interpreting you correctly here? – NikS Mar 29 '23 at 07:09
  • I've added some stuff that hopefully answers your questions, and (IMO) improves my answer. – Michael Weiss Mar 29 '23 at 16:09
  • @MichaelWeiss : OK, makes sense. BTW, it sounds like in the Platonist viewpoint, the $\omega$ of the “one true $V$” is generally understood to be the “standard $\omega$” (conforming to the “specific claim” of my prior comment)? That certainly would seem the most obvious choice for the “Platonic ideal” of $\omega$… – NikS Mar 30 '23 at 02:15
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    Yes, that’s right. – Michael Weiss Mar 30 '23 at 02:53