Partition models, Permutations of infinite sets without fixed points, and weak forms of AC

Abstract. In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice forms. • There does not exist an infinite Hausdorff space X such that every infinite subset of X contains an infinite compact subset. • If a field has an algebraic closure then it is unique up to isomorphism. • For every set X there is a set A such that there exists a choice function on the collection [A] 2 of two-element subsets of A and satisfying |X| ≤ |2 [A] 2 |. • Van Douwen’s Choice Principle (Every family X = {(Xi, ≤i) : i ∈ I} of linearly ordered sets isomorphic with (Z, ≤) has a choice function, where ≤ is the usual ordering on Z). We also extend the research works of B.B. Bruce [4]. Moreover, we prove that the principle “Any infinite locally finite connected graph has a spanning m-bush for any even integer m ≥ 4” is equivalent to K˝onig’s Lemma in ZF (i.e., the Zermelo–Fraenkel set theory without AC). We also give a new combinatorial proof to show that any infinite locally finite connected graph has a chromatic number if and only if K˝onig’s Lemma holds

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. • Plf,c (Every locally finite connected graph has a maximal independent set). • Plc,c (Every locally countable connected graph has a maximal independent set). • CACאα (If in a partially ordered set all antichains are finite and all chains have size אα, then the set has size אα) if אα is regular. • CWF (Every partially ordered set has a cofinal well-founded subset). • If G = (VG, EG) is a connected locally finite chordal graph, then there is an ordering <of VG such that {w < v : {w, v} ∈ EG} is a clique for each v ∈ VG

Free groups and the axiom of choice

The Nielsen–Schreier theorem states that subgroups of free groups are free. As all of its proofs use the Axiom of Choice, it is natural to ask whether the theorem is equivalent to the Axiom of Choice. Other questions arise in this context, such as whether the same is true for free abelian groups, and whether free groups have a notion of dimension in the absence of Choice. In chapters 1 and 2 we define basic concepts and introduce Fraenkel–Mostowski models. In chapter 3 the notion of dimension in free groups is investigated. We prove, without using the full Axiom of Choice, that all bases of a free group have the same cardinality. In contrast, a closely related statement is shown to be equivalent to the Axiom of Choice. Schreier graphs are used to prove the Nielsen–Schreier theorem in chapter 4. For later reference, we also classify Schreier graphs of (normal) subgroups of free groups. Chapter 5 starts with an analysis of the use of the Axiom of Choice in the proof of the Nielsen–Schreier theorem. Then we introduce representative functions – a tool for constructing choice functions from bases. They are used to deduce the finite Axiom of Choice from Nielsen–Schreier, and to prove the equivalence of a strong form of Nielsen–Schreier and the Axiom of Choice. Using Fraenkel–Mostowski models, we show that Nielsen–Schreier cannot be deduced from the Boolean Prime Ideal Theorem. Chapter 6 explores properties of free abelian groups that are similar to those considered in chapter 5. However, the commutative setting requires new ideas and different proofs. Using representative functions, we deduce the Axiom of Choice for pairs from the abelian version of the Nielsen–Schreier theorem. This implication is shown to be strict by proving that it doesn’t follow from the Boolean Prime Ideal Theorem. We end with a section on potential applications to vector spaces