# Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go through with merely ACC (countable choice). IS there a model of ZF+ACC where transfer actually fails for the definable hyperreal field of Kanovei and Shelah?

• Don't we expect models of ZF+ACC where the definition fails, simply by lack of ultrafilters? – Joel David Hamkins Jun 20 '16 at 13:19
• @JoelDavidHamkins, the definition is a formula in ZF. Otherwise one couldn't speak about a definable object at all. The magic trick is that in order to prove that the extension is actually proper you need more axiomatic material. In other words, models where there are no ultrafilters will simply give you the same field you started with.Notice that transfer happens to be true for the trivial extension :-) – Mikhail Katz Jun 20 '16 at 13:30

In this arxiv post (to appear in Journal of Symbolic Logic) we prove the existence of an explicitly definable set-theoretic construction in ZF (without if/else clauses) of a hyperreal extension ${}^\ast\mathbb R$. If in addition countable choice is assumed then we prove that the transfer principle holds for this $^{\ast}\mathbb R$. Assumping the existence of a free ultrafilter on $\mathbb N$ with well-orderable base (an assumption weaker than the well-ordering of $\mathbb R$) we show that the extension $\mathbb {R}\hookrightarrow{}^\ast\mathbb R$ is proper.