TITLE: Can ZFC prove "false theorems", and still be consistent? (was "Junk Theorems" follow up)
QUESTION [16 upvotes]: In Set theories without "junk" theorems? Jacques Carette consider's "junk theorems" of ZFC - theorems which are  artifacts of our means of encoding standard mathematical objects into set theory, but don't respect types.  For instance, given one encoding of the integers, it is a theorem that the only natural numbers which are functions are 0 and 3.  
The examples of junk theorems presented so far all immediately strike you as "junk" - basically there is a type error, and you can immediately recognizer that the junk theorem is the result of mixing types in a way you shouldn't.  The theorems are junk, not so much because they are wrong, but because they are not meaningful.
Is it possible for some false and meaningful statement in "normal mathematical language" to be a true statement of ZFC when it is unpackaged?
I am imagining some mundane mathematical question, like is the constant function 3 an element of the zero set of some functional F.  Unpackaging the set theoretic definitions of all of the terms involved shows that yes, 3 is a member of that set, but in fact F(3) is not zero.  Is this possible?
In other words, should we trust automated theorem provers?

REPLY [3 votes]: To Michael Blackmon (in comment on Can ZFC prove "false theorems", and still be consistent? (was "Junk Theorems" follow up) , sorry I don't have the cookie to add a new comment to that post, maybe someone with enough points can transfer this): 
Yes, the idea is that if there are two models of PA (one that affirms Goldbach's conjecture and one that refutes it), it's conceivable that ZFC disproves Goldbach's conjecture even though the conjecture is true in the standard integers.  That just means that the $\omega$ in every model of ZFC turns out to be nonstandard, i.e. ZFC itself is unsound (though still consistent) and proves theorems that are false for the standard integers.  This seems like an unlikely situation, but I don't see how it's nonsensical.  Am I missing something?