34 research outputs found
Symmetric Models, Singular Cardinal Patterns, and Indiscernibles
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). 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 approximation lemma. 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. 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 â”Ï. 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 Gitikâs 1980 paper âAll uncountable cardinals can be singularâ, a straightforward modification of which is in the last section of this chapter. That last section also tackles the question posed by Arthur Apter âWhich cardinals can become simultaneously the first measurable and first regular uncountable cardinal?â. Most of this last part is submitted for publication in a joint paper with Arthur Apter , Peter Koepke, and myself, entitled âThe first measurable and first regular cardinal can simultaneously be â”Ï+1, for any Ïâ. Throughout the chapter we show that several large cardinal axioms hold in the symmetric models we produce. 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. 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 Ï1-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 âblack boxesâ concerning the Dodd-Jensen core model are collected. 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
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
Benchmark Graphs for Practical Graph Isomorphism
The state-of-the-art solvers for the graph isomorphism problem can readily
solve generic instances with tens of thousands of vertices. Indeed, experiments
show that on inputs without particular combinatorial structure the algorithms
scale almost linearly. In fact, it is non-trivial to create challenging
instances for such solvers and the number of difficult benchmark graphs
available is quite limited. We describe a construction to efficiently generate
small instances for the graph isomorphism problem that are difficult or even
infeasible for said solvers. Up to this point the only other available
instances posing challenges for isomorphism solvers were certain incidence
structures of combinatorial objects (such as projective planes, Hadamard
matrices, Latin squares, etc.). Experiments show that starting from 1500
vertices our new instances are several orders of magnitude more difficult on
comparable input sizes. More importantly, our method is generic and efficient
in the sense that one can quickly create many isomorphism instances on a
desired number of vertices. In contrast to this, said combinatorial objects are
rare and difficult to generate and with the new construction it is possible to
generate an abundance of instances of arbitrary size. Our construction hinges
on the multipedes of Gurevich and Shelah and the Cai-F\"{u}rer-Immerman gadgets
that realize a certain abelian automorphism group and have repeatedly played a
role in the context of graph isomorphism. Exploring limits of such
constructions, we also explain that there are group theoretic obstructions to
generalizing the construction with non-abelian gadgets.Comment: 32 page
A Potpourri of Partition Properties
The cardinal characteristic inequality r <= hm3 is proved. Several partition relations for ordinals and one for countable scattered types are given. Moreover partition relations for lexicographically ordered sequences of zeros and ones are given in a no-choice context
Analytical Guide and updates for "Cardinal Arithmetic"
Part A: A revised version of the guide in "Cardinal Arithmetic" ([Sh:g]),
with corrections and expanded to include later works.
Part B: Corrections to [Sh:g].
Part C: Contains some revised proof and improved theorems.
Part D: Contains a list of relevant references