On Stone's theorem and the Axiom of Choice

It is a well established fact that in Zermelo-Fraenkel set theory, Tychonoff's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A. H. Stone's Theorem, that every metric space is paracompact, is considered here from this perspective. Stone's Theorem is shown not to be a theorem in ZF by a forcing argument. The construction also shows that Stone's Theorem cannot be proved by additionally assuming the Principle of Dependent Choice.

     

Kinna-Wagner selection principles, axioms of choice and multiple choice

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC).     

A parametrised choice principle and Martin's conjecture on Blackwell determinacy

We define a parametrised choice principle PCP which (under the assumption of the Axiom of Blackwell Determinacy) is equivalent to the Axiom of Determinacy. PCP describes the difference between these two axioms and could serve as a means of proving Martin's conjecture on the equivalence of these axioms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms.These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction.The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for.The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.     

Nielsen‐Schreier and the Axiom of Choice

The Nielsen‐Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen‐Schreier theorem implies the Axiom of Choice.     

Variants on the Berz sublinearity theorem

Aequationes mathematicae - We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice, which Berz used. We...     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Constructive Order Theory

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set‐theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well‐ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well‐ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.     

Projective-type axioms for the hyperbolic plane

It is shown that the Laws of Pappus and Desargues may replace the Axiom of Projectivities in Menger's development of hyperbolic geometry from axioms of alignment.     

The strength of Mac Lane set theory

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks.     

Closure operations that induce big Cohen–Macaulay algebras

We study closure operations over a local domain R that satisfy a set of axioms introduced by Geoffrey Dietz. The existence of a closure operation satisfying the axioms (called a Dietz closure) is equivalent to the existence of a big Cohen–Macaulay module for R. When R is complete and has characteristic $p>0$, tight closure and plus closure satisfy the axioms.We give an additional axiom (the Algebra Axiom), such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen–Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen–Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen–Macaulay algebra closure. This leads to proofs that in rings of characteristic $p>0$, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom.     

Disjoint Unions of Topological Spaces and Choice

We find properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice.     

Cofinalities of Linear Orders

We investigate whether the existence of long linear orders can be proved without the Axiom of Choice. This question has two different answers depending on its formalization.     

Filters,Antichains and Towers in Topological Spaces and the Axiom of Choice

We find some characterizations of the Axiom of Choice (AC) in terms of certain families of open sets in T₁ spaces.     

A New Proof that “Krull implies Zorn”

In the present note we give a direct deduction of the Axiom of Choice from the Maximal Ideal Theorem for commutative rings with unit. Mathematics Subject Classification: 03E25, 04A25, 13A15.     

Properties of the real line and weak forms of the Axiom of Choice

We investigate, within the framework of Zermelo‐Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SOME CONSEQUENCES OF THE TARSKI-KANTOROVITCH ORDERING THEOREM IN METRIC FIXED POINT THEORY

Abstract

We show that the Tarski-Kantorovitch Principle for continuous maps on a partially ordered set yields some fixed point theorems for contractive maps on a uniform space. Our proofs do not depend on the Axiom of Choice.     

Highway toll pricing

For a tolled highway where consecutive segments allow vehicles to enter and exit unrestrictedly, we propose a simple toll pricing method. It is shown that the method is the unique method satisfying the classical axioms of Additivity and Dummy in the cost sharing literature, and the axioms of Toll Upper Bound for Local Traffic and Routing-proofness. We also show that the toll pricing method is the only method satisfying Routing-proofness Axiom and Cost Recovery Axiom. The main axiom in the characterizations is Routing-proofness which says that no vehicle can reduce its toll charges by exiting and re-entering intermediately. In the special case when there is only one unit of traffic (vehicle) for each (feasible) pair of entrance and exit, we show that our toll pricing method is the Shapley value of an associated game to the problem. In the case when there is one unit of traffic entering at each entrance but they all exit at the last exit, our toll pricing method coincides with the well-known airport landing fee solution-the Sequential Equal Contribution rule of Littlechild and Owen (1973).     

On a problem of Woodin

A question of Woodin asks if is strongly compact and GCH holds for all cardinals , then must GCH hold everywhere. We get a negative answer to Woodin's question in the context of the negation of the Axiom of Choice. Received: 25 July 1997 / Revised version: August 6, 1998     

具有参数的一般的连续统假设与选择公理

<正> 本文解决了所有的具有参数为自:然数以及Aleph的一般的连续统假设GCH(u,v)之间以及与选择公理AC之间相互关系的问题.