# SET THEORY PDF FILE

This chapter introduces set theory, mathematical in- duction, and formalizes the notion of mathematical functions. The material is mostly elementary. For those of . A Book of Set Theory, first published by Dover Publications, Inc., in , is a .. when we have seen a few examples of these paradoxes; essentially, the. PDF Drive is your search engine for PDF files. As of today we have 78,, eBooks for you to download for free. No annoying ads, no download limits, enjoy .

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These notes for a graduate course in set theory are on their way to be- coming a book. .. Two important and well-known examples are: {v0: (¬(v0 = v0))}. SET THEORY. Set. A set is a collection of abstract objects. – Examples: prime numbers, domain in predicate logic. • Determined by (distinct) elements/members . Set theory is a basis of modern mathematics, and notions of set Examples: the set of students in this room; the English alphabet may be.

The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending "relativizing" it to make it more broadly applicable.

A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.

Fuzzy set theory[ edit ] Main article: Fuzzy set theory In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.

Inner model theory[ edit ] Main article: Inner model theory An inner model of Zermelo—Fraenkel set theory ZF is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. One reason that the study of inner models is of interest is that it can be used to prove consistency results.

For example, it can be shown that regardless of whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice , the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent has at least one model implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of determinacy and large cardinals , especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice.

Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy and thus not satisfying the axiom of choice. Many such properties are studied, including inaccessible cardinals , measurable cardinals , and many more.

These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory. Main article: Determinacy Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy.

The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property.

The axiom of determinacy AD is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved in particular, measurable and with the perfect set property. AD can be used to prove that the Wadge degrees have an elegant structure. Main article: Forcing mathematics Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails.

Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined i. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.

Main article: Cardinal invariant A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line.

These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory. Main article: Set-theoretic topology Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution.

Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question , a question in general topology that was the subject of intense research. We also study the related question of what can be said about a real in the ground model if it is recursive in generic reals of various sorts.

On the cofinality of ultrapowers, joint with Heike Mildenberger J. Symbolic Logic 64 PostScript or PDF We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g.

We also prove some related results for reduced powers with respect to filters other than ultrafilters. Bartoszynski and M. Scheepers Contemporary Math.

## Associated Data

The first aspect is to express the essential content of these proofs in a way that makes sense even in models where the inequalities hold trivially e. For this purpose, we use a Borel version of Vojtas's theory of generalized Galois-Tukey connections. The second aspect is to analyze a sequential structure often found in proofs of inequalities relating one characteristic to the minimum or maximum of two others. Vojtas's max-min diagram, abstracted from such situations, can be described in terms of a new, higher-type object in the category of generalized Galois-Tukey connections.

It turns out to occur also in other proofs of inequalities where no minimum or maximum is mentioned. I shall try to explain enough about each of them to be able to point out how they are connected.

## On Zermelo's and Von Neumann's Axioms for Set Theory

Since the underlying ideas of the two topics are quite different, I regard the existence of a connection as surprising. Rangaswamy and D. Hodges, J. Hyland, C. Steinhorn, and J. Truss Oxford Univ. In the downward-closed context, the ideal of meager sets is prime and b-complete where b is the bounding number , while the complementary filter is g-complete where g is the groupwise density cardinal.

We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results for measure in place of category are not provable in ZFC.

Girard, Y. Lafont, and L. Regnier London Math.

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Lecture Notes PostScript or PDF A category used by de Paiva to model linear logic also occurs in Vojtas's analysis of cardinal characteristics of the continuum. Its morphisms have been used in describing reductions between search problems in complexity theory. We describe this category and how it arises in these various contexts.

We also show how these contexts suggest certain new multiplicative connectives for linear logic. Perhaps the most interesting of these is a sequential composition suggested by the set-theoretic application.

Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-theoretic and set-theoretic. One of the set-theoretic properties and the associated cardinal are combinatorially natural, independently of any connection with algebra. Goebel, P. Hill, and W.

Liebert Contemporary Math.

London Math. Inspired by their methods, but using spaces of ultrafilters instead of their metric spaces, we prove a generalization of a theorem of Carlson about variable words. We extend this result to partitions of finite or infinite sequences of variable words, and we apply these extensions to strengthen a partition theorem of Furstenberg and Katznelson about combinatorial subspaces of the set of words.

As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.

Simple cardinal characteristics of the continuum Set Theory of the Reals ed. Judah Israel Math. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah's theorem that the dominating number is less than or equal to the independence number.

Ultrafilters related to Hindman's finite-unions theorem and its extensions Logic and Combinatorics ed.

Simpson Contemp. In particular, we study their relationship to various sorts of special ultrafilters on the natural numbers and to certain strengthenings of Hindman's finite-unions theorem due to Milliken and Taylor. The investigation also leads to some new strengthenings of Hindman's theorem.

I thank Peter Krautzberger for preparing a downloadable version of this old paper and the next two. Existence of bases implies the axiom of choice Axiomatic Set Theory ed. Baumgartner, D. Martin, and S. Shelah Contemp. Long ago, I started writing a book on ultrafilters.

## Set Theory Project

The project never got past Chapter 1, and even Chapter 1 existed only in handwritten form. This chapter contains basic, general information about ultrafilters. In the hope that it might be useful for some people, I had it scanned which had to be done manually, one page at a time, so I emphatically thank Robin Welshans who did the scanning , and I'm making it available here.The natural numbers as the least inductive set with the inclusion order.

Introduction to the foundations of geometry 6. We also show how these contexts suggest certain new multiplicative connectives for linear logic. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers. Its morphisms have been used in describing reductions between search problems in complexity theory. The field of effective descriptive set theory is between set theory and recursion theory.

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