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.

Inductive sets and the construction of the least inductive set. Induction proofs on the least inductive set. The axiom of Infinity. Models of Zermelo set theory. Chapter 2: Reconstruction of mathematics. The natural numbers as the least inductive set with the inclusion order. Inductive proofs of properties of the natural numbers: all natural numbers are transitive; all elements of natural numbers are natural numbers; every natural number consists of the set of all strictly smaller natural numbers; the natural numbers are totally ordered by inclusion.

The recursion theorem for functions on the natural numbers. Recursive definitions of addition and multiplication on the natural numbers via the Grassmann equalities. Definitions of the integers and the rational numbers as equivalence classes of pairs of simpler numbers.

Homework set 2. Third lecture. Chapter 2: Reconstruction of mathematics continued. Dedekind completion of a linear order. Definition of the real numbers. Completeness of the real numbers. Addition on the real numbers.

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Alternative constructions: Cauchy sequences or decimal expansions. Chapter 3: Further axioms of set theory. The Replacement Axiom Scheme. ZF0: Zermelo-Fraenkel without Foundation. Applications: iterated power sets and transitive closure. The sum of two orders; example: the sum of two copies of the natural numbers. The principle of order induction: the sum of two copies of the natural numbers satisfies the principle of order induction, but not the principle of complete induction.

Every wellorder satisfies the principle of order induction. Homework set 3. Fourth lecture. Chapter 3: Further axioms of set theory continued.

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Recursive constructions of a model of finite set theory and a model of Z in Zermelo-Fraenkel set theory; cf. Non-existence of loops. Axiom Scheme of Regularity. Wellorders are preserved by order sums and linear order products.

Examples of long well-orders and their embedding in the rational numbers.

Non-commutativity of sum and product operation for linear orders. Hartogs's Theorem without proof. Examples: natural numbers, the set of natural numbers; ordinals are closed under successor.

Homework set 4. Fifth lecture. Ordinals, Part II. Properties of ordinals: irreflexivity, elements of ordinals are ordinals, trichotomy law for ordinals. Initial segments.

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The Rigidity Theorem for Wellorders without proof ; cf. Homework exercise Isomorphic ordinals are equal. Representation theorem for wellorders: every wellorder is isomorphic to an ordinal.

Proof of Hartogs's Theorem. Existence of uncountable ordinals. Homework set 5. Sixth lecture. Induction and recursion on the ordinals. Transfinite induction and transfinite recursion. Addition, multiplication, and exponentiation of ordinals. Some properties of these operations: subtraction.

The von Neumann hierarchy. The axiom of regularity implies that every set lies in the von Neumann hierarchy proof sketch. Chapter 5: The Axiom of Choice. Choice functions. Existence of choice functions for finite sets. The Axiom of Choice. Zermelo's wellordering theorem. Proof of the wellordering theorem from the Axiom of Choice. Homework set 6. Seventh lecture. Chapter 5: The Axiom of Choice continued. Brief historical overview of the history of the independence of the Axiom of Choice.

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Fragments and Equivalents of the Axiom of Choice; examples without proof : "every vector space has a basis" equivalent , "there is a non-measurable set" proper fragment. Zermelo's wellordering theorem is an equivalent of the Axiom of Choice. Chapter 6: Cardinals. Cardinal numbers. Cardinal comparability theorem. Cardinal comparability is an equivalent of the Axiom of Choice. Homework exercises 31 and Successor and limit cardinals. Cofinal subsets of limit ordinals.

Regular and singular cardinals.

Successor cardinals are regular. Question about the existence of regular limit cardinals. The fundamental difference from classical set theory ZFC is that XST is based on a ternary membership relation, "x is an element of y with scope s.

Gao, S. Jackson, and Y. Zhang, A. Contemporary Math. We study several aspects of voting rules, with special attention to their relationship with ultrafilters. In particular, we study the set-theoretic hypothesis that all voting rules in the Boolean algebra of subsets of the natural numbers modulo finite sets are nearly ultrafilters.

We define the notion of support of a voting rule and use it to describe voting rules that are, in a sense, as different as possible from ultrafilters.

Finally, we consider how much of the axiom of choice is needed to guarantee the existence of voting rules. We determine all the provable implications between the four. We relate the associated cardinal invariants with bounding and dominating numbers and also with the uniformity of the meager ideal and some of its generalizations. Bagaria and S. Todorcevic, Trends in Mathematics, Birkhauser We obtain some additional information about such closed sets under certain assumptions involving the cardinal characteristics u and d.

Ultrafilters and Partial Products of Infinite Cyclic Groups, joint with Saharon Shelah PostScript or PDF We consider, for infinite cardinals k and m, the additive group of sequences of integers, of length k, with non-zero entries in fewer than m positions.

Our main result characterizes, in terms of the cardinals, when one such group can be embedded in another. The proof involves some set-theoretic results, one about families of finite sets and one about families of ultrafilters. Why sets? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far away planet.

Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science, in particular in database theory and formal methods.

Is there a good justification for that? We discuss these and some related issues. The weakenings involve 1 requiring intersections for only a fixed number of sets from the family and 2 requiring the sets to have elements near each other rather than actually intersecting.

These weakenings fit into a chain of implications, none of which are reversible under CH, but almost all of which are consistently reversible. We also connect these properties with weakened domination properties for families for functions on the natural numbers.

For unsplit families of sets, the chain of implications collapses from infinitely many properties to just four. Sums, Products, and Choice for Finite Sets PostScript or PDF We work in set theory without the axiom of choice, so infinite sums and products of cardinal numbers may not be well defined. We consider the special case of sums or products of countably many copies of the same finite cardinal number n.

We characterize the sets of integers that can, consistently with ZF, occur as the set of n such that the product or the sum of countably many copies of n is well defined. This paper is not intended for publication. Of the main results, one is due not to me but to Paul Howard, and the other is essentially already in my paper "Cohomology detects failures of the axiom of choice" Trans.

In many cases, there are weakenings of "adequate" that still imply that the family's cardinality is at least the characteristic in question. We analyze a few such weakenings. Our main results are partition theorems relating these weakenings to each other or to the original notions of adequacy. Among these are: If p,q has only two preimages, then both p and q are P-points, but, assuming the existence of at least 3 non-isomorphic, selective ultrafilters on N, there are p and q such that p,q has only 3 preimages but p is not a P-point.

If p,q has at most 5 preimages, then at least one of p and q is a P-point, but assuming the existence of at least 4 non-isomorphic selective ultrafilters on N, there are p and q such that neither of them is a P-point and p,q has only 6 preimages.

If p,p has at most 8 preimages then p is a P-point, but assuming the existence of infinitely many non-isomorphic, selective ultrafilters on N, there is a non-P-point p such that p,p has only 9 preimages. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size.

Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers containing 1 but not 0 , that a Dedekind-finite power of 2 cannot be divisible by 3, and that a Dedekind-finite set can be congruent modulo 3 to all of 0, 1, and 2 simultaneously.

In these results, 2 and 3 serve as typical examples; the full results are more general. We consider densities associated to ultrafilters on the natural numbers and investigate two additivity properties of such densities. Pure Appl. Call a set X of reals adequate for A if every real is A-related to one in X. Many cardinal characteristics of the continuum are defined as the smallest cardinality of any set adequate for a certain relation.

Call a real r needed for A if every set adequate for A has an element relative to which r is recursive. We study the reals needed for various relations A that arise in the theory of cardinal characteristics. 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.Cardinal comparability theorem. Pythagorean triples triples of integers a,b,c forming the sides of a right triangle, such as 3,4,5 Resolution of cubic equations Outer automorphisms of S 6. First-order Quantifiers Predicate Second-order Monadic predicate calculus.

The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. Applications[ edit ] Many mathematical concepts can be defined precisely using only set theoretic concepts. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St.

Seventh lecture.