Paul E. Howard

Limitations on the Fraenkel-Mostowski Method of Independence Proofs

The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorskis theorem.

Compactness in Countable Tychonoff Products and Choice

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

Disjoint Unions of Topological Spaces and Choice

We nd properties of topological spaces which are not shared by disjoint unions in the absence of some form of the Axiom of Choice. Introduction and Terminology This is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also (vd), (gt), (wgt), and (hkrr). Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions of topological spaces (See Denition 1, part 11.) For example, in set theory with choice the disjoint union of metrizable topological spaces is a metrizable topological space. The usual proof of this fact begins with the choice of metrics for the component spaces. We will show that the use of some form of choice cannot be avoided in this proof and in fact without choice the disjoint union of metrizable spaces may not even be metacompact. In section 1 we show that many assertions about disjoint unions of topological spaces are equivalent to the axiom of multiple choice. Models of set theory and corresponding independence results are described in section 2. In section 3, we study the roll the Axiom of Choice plays in the properties of disjoint unions of collectionwise Hausdor and collectionwise normal spaces. We begin with the denitions of the symbols and terms we will be using. Denition 1. 1. A familyK of subsets of a topological space (X;T )i sl.f .( locally nite) i each point of X has a neighborhood meeting a nite number of elements ofK. 2. X is paracompact i X is T2 and every open coverU of X has a l.f.o.r. (locally nite open renement)V.T hat is,V is a locally nite open cover ofX and every member ofV is included in a member ofU. 3. A familyK of subsets of X is p.f. (point nite) i each element of X belongs to only nitely many members ofK. 4. X is metacompact i each open coverU of X has an o.p.f.r. (open point nite renement). 5. An open coverU =fUi : i2 kg of X is shrinkable ithere exists an open cover V =fVi : i2 kg of non-empty sets such that V i Ui for all i2 k. V is also called a shrinking ofU. 6. X is a PFCS space i every p.f. open cover of X is shrinkable.

Non-constructive Properties of the Real Numbers

We study the relationship between various properties of the real numbers and weak choice principles.

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.

Versions of Normality and Some Weak Forms of the Axiom of Choice

We investigate the set theoretical strength of some properties of normality, in- cluding Urysohns Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of F subsets of normal spaces.

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).

Definitions of Compact

Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following: D efinition . A topological space X is 1. Compact (1) if every open cover of X has a finite subcover. 2. Compact (2) if every infinite subset E of X has a complete accumulation point (i.e., a point x 0 ∈ X such that for every neighborhood U of x 0 , | E ∩ U | = | E |). 3. Compact (3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover. 4. Compact (4) if each nest of closed, nonempty sets has a nonempty intersection. 5. Compact (5) if every family of closed sets in X which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection. 6. Compact (6) if each net in X has a cluster point. 7. Compact (7) if each net in X has a convergent subnet. This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (“the product of compact topological spaces is compact”) is equivalent to the Axiom of Choice, AC. Tychonoffs original proof that AC implies T used Definition 2 [13]. Other proofs have used Definitions 3 and 5; see [5]. The proof by Kelley that T implies AC uses Definition 5 [6].

The cardinal inequality α2 < 2α

Abstract In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set theory modulo AC). In this paper, we study the strength of the inequality α2 < 2α (in ZF, for every infinite cardinal number α, 2α ≰ α2; see [13]), where α is either the cardinality of special sets (see Definition 1 below) which are expressed as disjoint unions of finite, pairwise equipotent, sets lacking choice functions, or the cardinality of sets in specific permutation models of ZF0 set theory (i.e., ZF without the Axiom of Regularity).

Bases, spanning sets, and the axiom of choice

Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F, every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ℤ2 has a basis implies that every well-ordered collection of two-element sets has a choice function. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)