# An inner model and a model that contain same ordinals

In mathematical logic, suppose T is a theory in the language $L = \langle \in \rangle$ of set theory.

If $M$ is a model of $L$ describing a set theory and N is a class of M such that $\langle N, \in_M, \ldots \rangle$ is a model of T containing all ordinals of M then we say that N is an inner model of T (in M). Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension of V. (inner model, Wikipedia)

Question is, how can N contain all ordinals of M? Shouldn't a model contain ordinals as its domain? If models have same ordinals as its domain, aren't they basically the same, not needing any concept like inner model?

What am I getting wrong about a model? What makes an inner model different?

If I am not mistaken, it seems to say that ZFC has a countable inner model, which means countable domain - countable ordinals... and this just does not make sense.

• No, it does not say ZFC has a countable inner model. The requirement that the membership relation of $N$ is the restriction of the membership relation of $M$ is quite strong, and applying the Löwenheim–Skolem theorem generally does not produce models of this kind. – Zhen Lin Sep 18 '12 at 7:26

Now, to your question. There are two ways to look at universes of set theory. You can assume that you have one universe and you live inside this universe. Internally, this universe is not a set. We say that $M$ is an inner model of this universe if it is a transitive class containing all the ordinals, but has less sets, in which the axioms of ZF[C] hold. For example, you can define $L$ internally so you have $L$ as an inner model of every universe of ZFC.
We say, again, that $M$ is an inner model of $N$ if $M\subseteq N$, and $M$ contains all the ordinals of $N$, and so on. However between the ordinals of $N$ (the larger one) and the ordinals of the universe there doesn't have to be any relation whatsoever.
The domain of a set theoretical model does not just contain ordinals, it contains all the sets of the model, therefore two models with the same ordinals can differ in many ways. For example they can have different powersets $P(\alpha)$ of an ordinal $\alpha$, or the set of functions from an ordinal $\alpha$ to an ordinal $\beta$ can differ. This is the reason why the Continuum Hypothesis can not be decided within ZFC.