# Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ZFU are isomorphic in $INT_0$) of ZF and ZFU. What is the significance of this "synonymy" of ZF and ZFU given the meta-theoretic differences between ZFC and ZFCU (some of which were outlined in my question and the answers to it)? Does a similar "synonymy" claim hold between ZFC and ZFCU?

Does it show something about the significance of adding the Axiom of Choice? Does it give us reason to take the meta-theoretic differences "less seriously" (i.e., to think that future proofs will close this meta-theoretic gap between ZFC and ZFCU)?

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Do we know that ZFC and ZFCU are not synonymous? (As it happens I have that paper but haven't got round to reading it yet.) – Clive Newstead Jan 8 '13 at 20:56
@CliveNewstead Are you saying that you have a paper that shows ZFC and ZFCU are not synonymous? Or that you have the paper I reference in my post? If the former, I would be very interested in the paper, if you would be so kind as to point me towards it. – Dennis Jan 8 '13 at 21:02
Sadly not - I'm saying that I have (and intended to read, but didn't) the paper by Loewe and was asking if you knew that ZFC and ZFCU weren't synonymous. However Albert Visser has written a more general paper on synonymy, available here, in case you're interested. – Clive Newstead Jan 8 '13 at 21:06
@CliveNewstead Thanks for the reference, I don't know they aren't synonymous, my suspicion is merely a suspicion. In fact, I'll remove that from the question since it doesn't add anything. – Dennis Jan 8 '13 at 21:11
Does the proof that ZF and ZFU are synonymous carry over to ZFC and ZFCU? (Sorry, I realise I'm not being much help here.) – Clive Newstead Jan 8 '13 at 21:17