TITLE: Trouble with models of PA and ZFC
QUESTION [12 upvotes]: I have a big trouble in my mind, here is my false reasoning:

The Goodstein's theorem is undecidable in (first order) Peano Arithmetic.

There exist a non standard model N of PA where the Goodstein's theorem is false.

The Goodstein's theorem is provable in ZFC, using ordinals below 
$\epsilon_{0}$.

Since a statement provable in a theory T is true in all models of T, the Goodstein's theorem must be true in all models of ZFC. (?)
The ordinals of a ZFC model are well-founded from the internal point of view of this model.

If there exist a model M of ZFC where the model N is the naturals numbers of M. (?)

The Goodstein's theorem is false in M by definition of N, but true by 2), we can prove it in M using the 
$\epsilon_{0}$ of M, like in our universe.


Where I have gone wrong?

REPLY [7 votes]: The resolution to your confusion is simply that there is no model $M$ of $ZFC$ for which $N$ is the set of natural numbers.  Being the set of  natural numbers of a model of $ZFC$ is very rare for models of $PA$, because a set of natural numbers for a model of $ZFC$ has to satisfy all kinds of statements that $ZFC$ can prove about natural numbers and $PA$ cannot, whereas in many models of $PA$ some or all of those statements are false.
The contradiction that you demonstrated is actually a proof that there is no model of $ZFC$ which has $N$ as its to set of natural numbers.