A quick question about categoricity in model theory

Question

I just want to see if I am using the term "categoricity" correctly in the following context:

(1) I was thinking about why someone might reject a simple resolution to Skolem's Paradox.

(2) The resolution I had in mind was the following: model-theoretic truth is relative to what is in the model. Thus a set can be countable or uncountable -- it depends on the model. For example, it depends on whether a model is countable or uncountable.

(3) The rejection I had in mind: when we say that a model is countable, we are coming from a perspective of categoricity -- that is, a perspective from which we take countability to be absolute, or non-relative.

MY QUESTION: am I using "categoricity" right here?

Answer 1

Usually, one assume that your theory of set theory is consistent and hence there is a model $V$ of set theory. If you take the formal theory of set theory to be ZFC, there exists a set $\omega^V \in V$. A set of $V$ is said to be countable if there is a bijection $f \in V$ between $\omega^V$ and your set.

Now you can do model theory within $V$. That is, the domain of your structure is an element of $V$. In particular, you can develop the model theory of ZFC inside of $V$. By Downward-Lowenheim Skolem, ZFC has a countable model in $V$. That is there is set $G$ which is countable relative to $V$ and an interpretation function (also in $V$) that is a model of $ZFC$. $G$ is countable in $V$ means there is a $f \in V$ such that there is a bijection between $\omega^V$ and $G$.

However, $G$ itself is a model of set ZFC. So there is something in $G$ called $\omega^G$. There is also thing called $\aleph_1^G$. Since ZFC proves that $\aleph_1$ is not countable, one has that $\aleph_1^G$ is not countable in $G$. That is there is no $f \in G$ such that $f$ serves as a bijection between $\aleph_1^G$ and $\omega^G$.

However in the overlying universe $V$ $G$ is countable and hence all its subsets are countable. However the the bisection that witnesses this countability is not an element of model $G$.

So when you are working with several models of set theory where one model may be an element of another, then it would probably be more clear to distinguish countability in $V$ and countability in $G$. Hence a cardinal $\kappa$ will always refer to the cardinal $\kappa$ of $V$.

As for categoricity, I am not sure what you want to know relative to the idea of the skolem paradox.

A theory is $\kappa$-categorical if there is only one model of the theory up to isomorphism. However, this is assuming you fix a universe $V$ of set theory that serves as the "place" where all models of your theory comes from.

Answer 2

Let $\kappa$ be a cardinal. A theory $T$ is $\kappa$-categorical if any two models of $T$ of cardinality $\kappa$ are isomorphic.

A theory $T$ is categorical if any two models of $T$ are isomorphic. No (first-order) theory that has an infinite model is categorical.

As to the Skolem Paradox, the fact that a theory over a countable language, if it has an infinite model, has a countable model, is a nice result, with a quite accessible proof. A notable fact is that the result, for ZF, played a significant part in Cohen's proof that if ZF is consistent, then so is ZF plus the negation of the continuum hypothesis.

Answer 3

As Andre said, categoricity means that all models are isomorphic and $\kappa$-categoricity means that all models of size $\kappa$ are isomorphic.

The problem is that model theory is usually developed within a universe of ZFC. Things within think of that universe as a class, not as a set. If $\frak M$ is a countable model of ZFC living as a set inside a larger universe $V$, the model theory inside $\frak M$ does not have to coincide with the model theory inside $V$.

In fact this is true for class models, that is $V$ can be a "very large" model of ZFC, and there is a subclass of $V$ which is also a model of ZFC that is significantly "smaller" and these two models have different model theories (because they ultimately contain different sets).

When we say that a model of ZFC is countable, we come from an external point of view. That is we live in a very large universe and that universe happened to know a countable set $M$ with a binary relation $E$ such that $\mathfrak M=\langle M,E\rangle$ is a model of ZFC.

It is possible that $M$ has a subset $N$ such that $\mathfrak N=\langle N,E\cap(N\times N)\rangle$ is also a model of ZFC, and it is possible that these models are not isomorphic at all.

This tells you that ZFC is not $\aleph_0$-categorical, and that categoricity here is far from categoricity in philosophy.

Furthermore, Cohen's work on forcing shows us that countability is far from absolute. If $\frak M$ is a countable model of ZFC, then there is a "slightly larger" $\frak A$ which is a countable model of ZFC, but $\frak A$ thinks that $\omega_1^\frak M$ is countable. Namely we took a set that $\frak M$ thought is uncountable, and we added bijections between this set and $\omega$.

However, as I said before we do all these model theoretic considerations inside a fixed universe of ZFC (this universe is not a set in our context), and the question whether or not a set is countable is answered in this universe. The notion of countability, therefore, does not "go down" to models which live inside this universe. They have their own notion of countability.