Countability of models of ZFC

Question

I'm reading about Skolem's paradox and the idea that countability in set theory is "relative to a model", and that makes we wonder what is actually meant by saying that a model of ZFC is countable or not.

Suppose for example that $\mathcal M_0$ is an uncountable model of ZFC. If I understand correctly, what this means is that $\mathcal M_0$ is itself a set in another model, say $\mathcal M_1$, such that $\mathcal M_0$ is an uncountable set in $\mathcal M_1$ (For example, there might exist inside $\mathcal M_1$ a bijection between $\mathcal M_0$ and the power set of $\mathbb N $). But now $\mathcal M_1$ could be a set inside yet another model $\mathcal M_2$, and it might be countable there, since countable models exist. If this is the case then inside $\mathcal M_2$, $\mathcal M_0 \subset \mathcal M_1$ is a subset of a countable set and is therefore countable itself. So $\mathcal M_0$ is uncountable in $\mathcal M_1$ but countable in $\mathcal M_2$.

Can such a scenario exists ? if so is there an absolute meaning to the statement that a model is uncountable ?

Answer 1

Sure, this can happen. For instance if $M_2$ is a model of "ZFC + there exist transitive models of 'ZFC + there is an inacessible cardinal'", then in $M_2$, there is a countable transitive model $M_1$ of 'ZFC + there is an inacessible cardinal', and if we let $\kappa$ be an inaccessible in $M_1$, then $M_0 = V_\kappa^{M_1}$ is a model of ZFC that is uncountable with respect to $M_1$ and countable with respect to $M_2.$

When we talk about models of ZFC, or any mathematical subject, we are usually doing so relative to a fixed background. For instance we are working in some set theory (ZFC plus enough large cardinals, say), and saying that a given set is countable is a statement with absolute meaning. So this goes for models of ZFC too. A model of ZFC is a set $M$ and a relation $E$ on $M$ such that $(M,E)$ satisfies all of the ZFC axioms. It's an uncountable model when $M$ is an uncountable set. That's it. It's no different from saying $(\mathbb R, <)$ is an uncountable linear order whereas $(\mathbb Q, <)$ is a countable linear order.

However, you could also think of working in a "fixed background" (i.e. a "fixed set theoretical universe") as working inside a "fixed model" of ZFC.$^*$ And perhaps that fixed model could be a set, perhaps even a countable set, in some larger universe. Now we get into philosophy. Is there absolute meaning to any mathematical statement? Certainly if we believe there is one "real" set theoretical universe, we could always assume it is the background. But there are different ways to think about these things.

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$^*$ e.g. in the example of my first paragaph, we could have instead said "assume there are transitive models of 'ZFC + there is an inacessible cardinal'" and never mentioned the model $M_2,$ thinking of it as the set theoretical universe instead, and we'd just have an example of a model $M_0$ that is an uncountable model according to some larger model $M_1$, but is actually countable. As it stands, we assumed $M_2$ was some set that satisfied this theory to get an example of what you asked for in the question. We didn't specify what size $M_2$ was. We could have assumed it was countable or any size, but we didn't stipulate anything since it wasn't relevant.

Answer 2

This topic is really in the philosophy of set theory and is very well studied. Rather than repeat the literature, I would recommend this article as a starting point:

Hamkins, Joel David (2012). The set-theoretic multiverse. The Review of Symbolic Logic 5(3), 416–49. https://doi.org/10.1017/S1755020311000359. ArXiV: https://arxiv.org/abs/1108.4223

Abstract The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

Another way of phrasing this issue, which set theorists sometimes use, is to ask whether set theory is the study of sets (the "universe view") or the study of models of set theory (related to the multiverse view). Hamkins' paper is clearly on the multiverse side; there are a number of set theorists who argue equally strongly on the universe side.

Another relevant paper is

Feferman, Solomon, Harvey M. Friedman, Penelope Maddy, and John R. Steel (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6(4), 401–46. https://doi.org/10.2307/420965.

As is seen in these articles, set theorists are less interested in the question of whether every set is really countable, but very interested in whether questions like the continuum hypothesis have well-defined, objective truth values. For CH, of course, there is an enormous literature and great understanding of models in which CH is true and models where it is false.

Answer 3

There are (at least) two (philosophical) approaches to set theory: Axiomatic one (syntactic leaning/formalist) and Platonic one (semantic leaning/realist). When we say $\mathcal M_0$ is an uncountable model of ZFC, we mean $\mathcal M_0$ is genuinely a Platonic existence of a set that is uncountable (such as $\mathbb R$), where uncountability has semantic meaning instead of syntactic one.

It's best to think of this as how one understands countability/cardinality/(naive) set theory in general when one first studies them before formal logic. That is, given a set, it's either countable or not, there is no ambiguity. However, if we consider a set $S\in \mathcal M_0$, whether it's countable or not relative to the model has a different interpretation. For example, $\mathcal M_0$ itself as a model must be a set, but in its own interpretation of ZFC, it's a proper class, not a set.

Now one may argue this is insane, as the Platonic universe of sets may not exist and even if it does (for which, every proposition such as CH must have a definite truth value, which we couldn't know or agree with each other about), it cannot be understood by our finite minds. The only approach to study the Platonic universe is to set up an effective axiomatic framework to reason about sets. This is totally a valid objection, and as far as I know, this has no satisfying answer. Your argument is essentially taking this approach to say there is no the ultimate Platonic entity of sets, but only models of axioms (though philosophically this is also problematic: what exactly is a set without Plato? Maybe this is a more complicated version of actual vs potential infinity... anyway.), therefore your $\mathcal M_0$ always needs to be interpreted in another model. But in this approach, whether a model is countable or not is a meaningless question, because of the infinite hierarchy, all we have are relative notions: when you say $\mathcal M_0$ is countable or not, you have to specify in which sense/model you're using the word "countable".

Answer 4

One can very well believe that there is absolutely no such thing as any truly uncountable collection in terms of size, while believing that there are uncountable collections in terms of complexity. This might seem crazy, but it is not. If you assume Z set theory (or much less), then indeed there is only one notion of countable, because the other notions are equivalent to it. But if you use a different foundation, then there might be at least two distinct notions:

(1) Countable: S is countable iff there is an injection from S to ℕ.
(2) Weak-Enumerable: S is weak-enumerable iff there is a partial function on ℕ whose range contains S.

(1) expresses countability in terms of complexity, because S may be so complicated that there is no way of distinguishing members of S in order to have an injection from S to ℕ.

(2) expresses countability in terms of size, because if S is truly no bigger than ℕ then we can weakly enumerate it in the sense of adding each member x of S to our enumeration whenever we figure out that x is indeed a member of S.

Again, note that (2) implies (1) in typical set theories, because of their way of expressing "partial function". But it may not in other foundations!

In a foundation with intrinsic partial functions, we might be able to prove that ℕ→ℕ is not countable (1) but be unable to prove that it is not weak-enumerable (2), and moreover the intended model for that foundation may very well be weak-enumerable! The reason is that, if ℕ→ℕ is weak-enumerable witnessed by a partial function F, then we cannot construct an injection G from ℕ→ℕ to ℕ by defining G(h) to be the minimum k∈ℕ such that F(k) = h or 0 if no such k exists, since "∃k∈ℕ ( F(k) = h )" is not boolean and so G would be merely a partial function.

In this foundation, Skolem's paradox is practically gone, because the paradox arises from the idea that uncountable collections are strictly bigger than countable collections. But as mentioned above, ℕ→ℕ can be smaller than ℕ (2) but still not countable (1). Although we should not assume that the world is weak-enumerable, it would be consistent to do so, and in that viewpoint every collection (including every model of ZFC) would not be larger than ℕ. So it is completely unsurprising that a model M of ZFC can be countable; it just means that M is simple enough that there is an injection from the domain of M to ℕ! If M is too complicated, then M is not countable. But in this viewpoint it is always the case that M is not larger than ℕ!

Many of the above points also apply to analysis of Skolem's paradox from a conventional set-theoretic viewpoint. But I mention this viewpoint because you need to realize that your notion of countability is very much dependent on your foundations, and splits into at least two notions in some foundations. In particular, (1) is certainly not absolute. But if your foundational meta-system MS is like above plus an axiom that every type (including the universal type) is weak-enumerable, then MS trivially can prove that every model of MS is weak-enumerable, and every model itself also thinks that its universe is weak-enumerable. Whether you believe this kind of MS or not, countability in the form of (2) is absolute in it!