Why does $\omega$ have the same cardinality in every (transitive) model of ZF?

Solution 1:

Suppose $X$ and $Y$ are transitive models of set theory. Then the elements of $X$ and $Y$ are sets, and the set membership relations $\in^X$ and $\in^Y$ are the normal set membership relation $\in$.

Hence we can build up the following sequence of results, using the assumption of transitivity:

• $\emptyset^X$ (the unique element $z$ of $X$ such that $X \models (\forall y) [y \not \in z]$) and $\emptyset^Y$ are both the real empty set, $\emptyset$.

• So $1^X = \{\emptyset^X\}$ and $1^Y$ are both $1 = \{ \emptyset\}$.

• Similarly, for each finite ordinal $n$, $n^X = n^Y = n$.

• Moreover, for any $z \in X$, if $X$ satisfies the sentence "z is a finite ordinal", then $z$ really is a finite ordinal. The same holds for $Y$.

• Thus $\omega^X = \omega = \omega^Y$. So not only do $\omega^X$ and $\omega^Y$ have the same cardinality, they are literally the same set.

Solution 2:

If a model is transitive then its ordinals are real ordinals. In particular it means that any finite ordinal in the model is actually finite, and therefore $\omega$ is the real $\omega$.

It is important to note, however, that there is a possibility that countable sets are not countable in the model; and that it is possible for non-transitive (and non-well founded) models to exist in which there is an uncountable number of finite ordinals, of course as Carl points out, internally the finite ordinals are always a countable set. It is only when we look at the model from an external point of view we can tell it has an uncountable number of finite sets.

The important point to notice is that we often assume that there is an "absolute" universe in which we do mathematics, and this universe is a model of ZFC. While this model is not a set, when we talk about models of ZFC we talk about set models. Now there is an internal and external issues to discuss. It is always the case with set models that there is an element of the set-model which the set model "sees" as having a very big cardinality, but the universe sees as having a "small" cardinality.

For example, if $M$ is a countable transitive model then the real numbers of $M$ is a countable set, but not internally.

For non-transitive models the natural numbers could be uncountable as well, externally, at least. However for a transitive model this is impossible.