Trouble with models of PA and ZFC 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?
 A: 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.
A: As Andrés points out in the comments, your question seems to be resolved by the observation that not every model of PA can arise as the $\mathbb{N}$ of a model of ZFC. 
Meanwhile, one can attempt to understand more deeply exactly which models of PA do arise as the $\mathbb{N}$ of a model of set theory. Let us say that a model $N\models\text{PA}$ is a standard model of arithmetic (as opposed to the standard model of arithmetic), or alternatively is a ZFC-standard model of arithmetic, if $N=\mathbb{N}^M$ for some $M\models\text{ZFC}$. 
Thus, a model of arithmetic is a standard model of arithmetic, if it is the standard model of arithmetic from the perspective of some model of set theory. These are the models of PA that seem to be at the heart of your question.
We can characterize exactly what these models are as follows.
Theorem. (Ali Enayat) The following are equivalent, for a countable nonstandard model $N\models\text{PA}$. 


*

*$N$ arises as $\mathbb{N}^M$ for some model $M\models\text{ZFC}$.

*$N$ is computably saturated and satisfies all the arithmetic consequences of ZFC. 
In statement 2, what we mean is that $N\models\varphi$, whenever $\varphi$ is an arithmetic statement that is provable from ZFC. For example, this includes the case where $\varphi$ is Goodstein's theorem, and this is exactly what is going on in your question. Obviously, any model of PA arising as the $\mathbb{N}$ of a model of ZFC set theory must satisfy all the arithmetic consequences of ZFC, and this is essentially how you are reasoning in statement 3 of the question. The interesting part is that this is also sufficient, when combined with computable saturation. 
You can find a proof of the theorem in my paper: J. D. Hamkins and R. Yang, Satisfaction is not absolute, to appear in the Review of Symbolic Logic, pp. 1-34. (arxiv:1312.0670) (see proposition 3). 
