Łos’s theorem for (bounded) D-ultrapowers, D being the

ultrafilter introduced by Kanovei and Shelah [Journal of Symbolic Logic,

69(1):159–164, 2004], can be established within Zermelo–Fraenkel set

theory plus Countable Choice ($ZF+AC_\omega$). Thus, the Transfer Principle

for both Kanovei and Shelah’s definable nonstandard model of the reals

and Herzberg’s definable nonstandard enlargement of the superstructure

over the reals [Mathematical Logic Quarterly, 54(2):167–175; 54(6):666–

667, 2008] can be shown in $ZF+AC_\omega$. This establishes a conjecture by

Mikhail Katz [personal communication].