The Wikipedia page on true arithmetic says that it has $2^\kappa$ models for each uncountable cardinal $\kappa$. This refers to the theory of all first-order statements of the naturals.
I'm curious how CH affects this (if it does at all). With the hyperreals, for instance, if we have CH, all of the various choices of ultrafilter on countable sequences of reals produce isomorphic structures. Do we get the same thing with the hypernaturals?
Basically, the questions are the following:
- Without CH, how many $\mathfrak c$-sized models of the first-order theory of the naturals are there? And the reals?
- With CH, how many models of both are there?
- Does CH change how many different models exist, or does it only change how many different models are obtainable using the ultrapower construction?
- When that Wikipedia page says "true arithmetic" has $2^\mathfrak{c}$ models of cardinality $\mathfrak{c}$, is there any implicit assumption being made about the status of CH?