# Continuum hypothesis in nonstandard universe

In Vladimir Kanovei's book "Nonstandard Analysis, Axiomatically", some nonstandard set theory is introduced. It seems that, one of them, DNST, is useful.

When we are talking about higher order objects, usually we need GCH. I want to know that, is GCH consistent to DNST.

There are two kinds of GCH, one is the external version, and another is the standard version $$GCH^{st}$$.

We just need to prove the consistence for the subsystem $$HST'_\kappa+GCH+GCH^{st}$$, for every definable $$\kappa$$, and then use the compactness theorem.

It seems that the standard version of GCH is easy. But I don't know how to prove the consistence of the external version.

• I'll have to check to be sure, but I had thought that the consistency proofs proceed by iterated ultrapowers of a model of ZFC along a linear order. In this case, if you start with a model of ZFC+GCH, you will get GCH at every level of nonstandardness. – Joel David Hamkins Nov 21 at 15:47