“$\fomom$中的极大链和整数的超幂”的附录

{}^ωω中的极大链与整数的超幂

Maximal Chains in {}^ωωand Ultrapowers of the Integers

各种问题所提出的P. Nyikos有关超过滤器的$ω$和链的偏序$(ω,<^*)$回答。主要工具是Oracle链条件及其变体。

Isomorphic limit ultrapowers for infinitary logic

The logic L^1_θintroduced in [Sh:797]; it is the maximal logic below L_theta theta in which a well ordering is not definable. We investigate it for theta a compact cardinal. We prove it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are L^1_theta-equivalent iff for some omega-sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.

An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On-saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG. In NBG, it can be…