Symmetric Models, Singular Cardinal Patterns, and Indiscernibles

Abstract

This thesis is on the topic of set theory and in particular large cardinal axioms, singular cardinal patterns, and model theoretic principles in models of set theory without the axiom of choice (ZF).<br /> The first task is to establish a standardised setup for the technique of symmetric forcing, our main tool. This is handled in Chapter 1. Except just translating the method in terms of the forcing method we use, we expand the technique with new definitions for properties of its building blocks, that help us easily create symmetric models with a very nice property, i.e., models that satisfy the <em>approximation lemma</em>. Sets of ordinals in symmetric models with this property are included in some model of set theory with the axiom of choice (ZFC), a fact that enables us to partly use previous knowledge about models of ZFC in our proofs. After the methods are established, some examples are provided, of constructions whose ideas will be used later in the thesis. <br /> The first main question of this thesis comes at Chapter 2 and it concerns patterns of singular cardinals in ZF, also in connection with large cardinal axioms. When we do assume the axiom of choice, every successor cardinal is regular and only certain limit cardinals are singular, such as ℵ_ω. <br /> Here we show how to construct several patterns of singular and regular cardinals in ZF. Since the partial orders that are used for the constructions of Chapter 1 cannot be used to construct successive singular cardinals, we start by presenting some partial orders that will help us achieve such combinations. The techniques used here are inspired from Moti Gitik’s 1980 paper “All uncountable cardinals can be singular”, a straightforward modification of which is in the last section of this chapter. That last section also tackles the question posed by Arthur Apter “Which cardinals can become simultaneously the first measurable and first regular uncountable cardinal?”. Most of this last part is submitted for publication in a joint paper with Arthur Apter , Peter Koepke, and myself, entitled “The first measurable and first regular cardinal can simultaneously be ℵ_{<em>ρ</em>+1}, for any <em>ρ</em>”. Throughout the chapter we show that several large cardinal axioms hold in the symmetric models we produce. <br /> The second main question of this thesis is in Chapter 3 and it concerns the consistency strength of model theoretic principles for cardinals in models of ZF, in connection with large cardinal axioms in models of ZFC. The model theoretic principles we study are variations of Chang conjectures, which, when looked at in models of set theory with choice, have very large consistency strength or are even inconsistent. <br /> We found that by removing the axiom of choice their consistency strength is weakened, so they become easier to study. Inspired by the proof of the equiconsistency of the existence of the ω₁-Erdös cardinal with the original Chang conjecture, we prove equiconsistencies for some variants of Chang conjectures in models of ZF with various forms of Erdös cardinals in models of ZFC. Such equiconsistency results are achieved on the one direction with symmetric forcing techniques found in Chapter 1, and on the converse direction with careful applications of theorems from core model theory. For this reason, this chapter also contains a section where the most useful ‘black boxes’ concerning the Dodd-Jensen core model are collected. <br /> More detailed summaries of the contents of this thesis can be found in the beginnings of Chapters 1, 2, and 3, and in the conclusions, Chapter 4.