Assume (M,∊M) is a model of ZF. Assume also that (n,∊n) ∊ M is a model in the sense of M and (N,∊N) is a model in the real world with the property that for all sentences σ


N ⊨ σ   if and only if   M ⊨ (n ⊨ σ).


By using conjunction ∧ it follows that, if T is a finite set of sentences then


N ⊨ T   if and only if   M ⊨ (n ⊨ T).   (1)


For example, if T consists of only two sentences σ1 and σ2 then


N ⊨ T   iff   N ⊨ σ1∧σ2   iff   M ⊨ (n ⊨ σ1∧σ2)   iff   M ⊨ (n ⊨ T) 


However, for an infinite set of sentences T the same argument to prove (1) does not work. Furthermore I think that (1) is not true for infinite T but the reasons for this are unclear to me.


Let's take T=ZF for example. To prove (1), could we not proceed as follows? First we construct ZF (the set of sentences) inside M. We can surely do this inside a model of ZF. Let's call this set ZFM. Then if we knew that there is some kind of correspondence between ZFM and the real world ZF, then using this correspondence, could we somehow deduce (1) or even one implication of the equivalence (1)?


My question is that what kind of relationship (if any) there is between the set of sentences constructed in the model M, say TM, and the real T. And if they happen to be fundamentally different, then what are the reasons behind this? Furthermore, is there some property of the model M which guarantees that TM and the real T are essentially the same?



Models within a model of set theory
We will always have ${\rm ZF}^M\supseteq {\rm ZF}$. The problem comes
if $M$ contains nonstandard sentences--which is equivalent to the question of whether $M$ contains nonstandard natural numbers since we can identify formulas with their Godel codes.
If there are no nonstandard integers, then they type of externl induction you are doing works fine, but if there aren't it might break down at a nonstandard stage.


