Maximal Chains in {}^ωωand Ultrapowers of the Integers

Addendum to ``Maximal Chains in $\fomom$ and Ultrapowers of the Integers''

Upon presenting the proof of Theorem 3.3 in "Maximal chains in $$ and ultrapowers of the integers" I discovered that it is not entirely correct and certainly some details should be added. I have therefore written an addendum to the paper and made it available by ftp. Unfortunately the published version will be somewhat wanting.

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…

关于几乎有界结构的超滤扩张的注记

Notes on ultrafilter extensions of almost bounded structures

本文针对度数无全局有限上界或存在无穷度数元素的结构研究难点，提出几乎有界结构概念，证明其仍为超滤扩张的初等子结构、初等嵌入可提升且可数情形下扩张与特定超幂同构，并评述相关模态逻辑。

Saturating the random graph with an independent family of small range

Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = λ> \aleph_0$, the fact that $P(I)/\de$ has little freedom (as measured by the fact that any maximal antichain is of size $<λ$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that $M^I/\de_1$ is $λ^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true…

Constructing regular ultrafilters from a model-theoretic point of view

This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected…