Hi, I would like some help on something that in some ways has already been touched in the past here. I saw these relevant questions being answered, but I am still unable to understand some things. I would be grateful if someone could help me.


Let $M \models$ ZFC that is countable and let $p(x) = {0 \in x, \ldots, n \in x, \ldots} \cup {x \in \omega }$ be a type where $0, \ldots, n, \dots$ are the finite ordinals of $M$, and $\omega$ the first limit ordinal of $M$.
Now let $N$ be an $\omega\_1$-saturated model s.t. $M \prec N$. Now $p(x)$ obviously is fin. satisfiable in $M$, therefore it is satisfiable in $N$, by let's say some set $a$.
Both $M$ and $N$ are ZFC models, therefore natural numbers and $\omega$ are absolute for $M, N$. So how is it possible to have an ordinal $a$ (since $a \in \omega$) which is finite (in $N$ and therefore in $M$) but different from all $0, \dots, n, \ldots$?


In the same way, given any countable set $b$ of finite sets of $M$, I can always find a saturated model $N$ s.t. $M \prec N$, with some $c \in\_N b$ and $c$ infinite. Is this correct?



Saturated extensions of ZFC models
I'm addressing both the question and the comments, but possibly this question should be closed. First let's be clear what we mean by a model of set theory. A model is a set $M$ (or perhaps a class) with a binary relation $E$. $(M,E)$ satisfies *Foundation* if for every $x$ in $M$, there is an $E$-minimal $y$
in $M$ such that $y E x$. $M$ is *well founded* if for every $x \subseteq M$, there is an $E$-minimal $y$ in $M$ such that $y$ is in $x$. If $E$ is $\in$, then $M$ is *transitive* if every element of $M$ is a subset of $M$. Well foundedness is implicit in transitivity.


The problem is that your model need not be well founded. The Axiom of Foundation only implies that the model of set theory you are working with does not have an element witnessing that it is ill founded. But certainly such a witness can exist outside. The Henkin Construction essentially never will result in a well founded model. A simpler example is given by a non standard model of PA. The model satisfies the induction scheme (which is the analog here of Foundation) but any well founded model of PA is isomorphic to the standard model.


Furthermore, any well founded model $(M,E)$ of ZFC is isomorphic to a transitive set (or class) with
the membership relation. This is the Mostowski collapse at work. So transitivity is not the issue here. The failure of finiteness to be absolute is tied up in the ill foundedness of the model in question.


