Bases in vector spaces and the axiom of choice

It is shown that the axiom of choice follows in a weaker form than the Zermelo - Fraenkel set theory from the assertion: every set of generators G of a vector space V over the two element field includes a basis L for V. It is also shown that: for every family A = { A i : i ∈ k } \mathcal {A}=\{A_i:i\in k\} of non empty sets there exists a family F = { F i : i ∈ k } \mathcal {F=}\{F_i:i\in k\} of odd sized sets such that, for every i ∈ k i\in k , F i ⊆ A F_i\subseteq A iff in every vector space B B over the two-element field every subspace V ⊆ B V\subseteq B has a complementary subspace S S iff every quotient group of an abelian group each of whose elements has order 2 has a set of representatives.

On ultracompact spaces in ZF

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

The so called Generalized Continuum Hypothesis (GCH) is the sentence: "If A is an infinile set whose cardinal number is K and 2K denotes the cardinal number of the set P(A) of subsets of A (the power set of A), and K + denotes the succesor cardinal of K, then 2K = K +". The Continuum Hypothesis (CH) asserts the particular case K = o. It is clear that GCH implies CH. Another equivalent version of GCH, is the sentence: 'Any subset of the set of subsets of a given infinite set is or of cardinality less or equal than the cardinality of the given set, or of the cardinality of all the set of subsets". Gödel in 1939, and Cohen in 1963, settled the relative consistency of the Axiom of Choice (AC) and of its negation not-AC, respectively, with respecllo the Zermelo-Fraenkel set theory (ZF). On the other hand, Gödel in 1939, and Cohen in 1963 settled too, the relative consistency of GCH , CH and of its negations not-GCH, not-CH, respectively, with respect to the Zermelo-Fraenkel set theory with the Axiom of Choice (ZF + AC or ZFC). From these results we know that GCH and AC are undecidable sentences in ZF set theory and indeed, the most famous undecidable sentences in ZF; but, which is the relation between them? From the above results, in the theory ZF + AC is not demonstrated GCH; it is clear then that AC doesn't imply GCH in ZF theory, Bul does GCH implies AC in ZF theory? The answer is yes! or equivalently, there is no model of ZF +GCH + not-AC. A very easy proof can be given if we have an adecuate definition of cardinal number of a set, that doesn't depend of AC but depending from the Regularity Axiom, which asserls that aIl sets have a range, which is an ordinal number associated with its constructive complexity. We define the cardinal number of A, denoted |A|, as foIlows: |A|= { The least ordinal number equipotent with A, if A is well orderable The set of all sets equipotent with A and of minimum range, in other case. It is clear that without AC, may be not ordinal cardinals and all cardinals are ordinal cardinals if all sets are well orderable (AC). Now we formulate: GCH*: For all ordinal cardinal I<, 2K = I< + In the paper is demonstrated that this formulation GCH* is implied by the traditional one, and indeed equivalent to it. Lemma, The power set of any well orderable set is well orderable if and only if AC. This is one of the many equivalents of AC in ZF,due lo Rubin, 1960. Proposition. In ZF is a theorem: GCH* implies AC. Supose GCH*. Let A be a well orderable set; then |A| = K an ordinal cardinal, so A is equipotent with K and then P~A) is equipotent with P(K); therefore |P(A)I|= |P(K)| = 2K = K+. But then |P(A)|= K+ and P(A) 'is equipotent with K+ and K+ is an ordinal cardinal; therefore P(A) is well orderable with the well order induced by means of the bijection, from the well order of K+. Corolary: In ZF are theorems: GCH impIies AC and GCH is equivalent to GCH*. We see from this proof, that GCH asserts that the cardinal number of the power set of a well orderable set A is an ordinal, which is equivalent to AC, but GCH asserts also that that ordinal cardinal is |A|+ , the ordinal cardinal succesor of the ordinal cardinal of the well orderable set A.

Bases in Vector Spaces and the Axiom of Choice

On first and second countable spaces and the axiom of 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.

The Baire Category Theorem and choice

We work in set theory without the axiom of choice: ZF . We show that the axiom BC : Compact Hausdorff spaces are Baire , is equivalent to the following axiom: Every tree has a subtree whose levels are finite , which was introduced by Blass (cf. [ 4 ]). This settles a question raised by Brunner (cf. [ 9 , p. 438]). We also show that the axiom of Dependent Choices is equivalent to the axiom: In a Hausdorff locally convex topological vector space, convex-compact convex sets are Baire . Here convex-compact is the notion which was introduced by Luxemburg (cf. [ 16 ]).

About the Axiom of Choice

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ? of natural numbers such that for everyn ? ?, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x ? y, but no one-to-one map h : y ? x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo---Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether "every Dedekind-finite set is finite" implies NDS addressed in G. H. Moore "Zermelo's Axiom of Choice. Its Origins, Development, and Influence" and in P. Howard---J. E. Rubin "Consequences of the Axiom of Choice". We also prove that AC WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, "every Dedekind-finite set is finite" ? NDS in ZF, either) and that "for all infinite cardinals m, m + m = m" ? NDS in ZFA.

E. Specker has proved that the axiom of choice (AC) is false in NF [6]. Since AC is stratified, one can, according to another famous result of Specker [7], prove directly ¬AC in type theory (TT) plus some finite set of ambiguity axioms, i.e. sentences of the form φ ↔ φ+, where φ+ results from φ by adding one to its type indices.We shall in §2 of this paper give a disproof of AC directly in TT plus some axioms of ambiguity. The argument will be split into two parts. The first one (contained in Proposition 2) concerns cardinal arithmetic and has nothing to do with typical ambiguity. Though carried out in TT, it could have been done in other set theories such as Zermelo's Z or ZF. The second part is an application of this to the cardinals of the universes at different types. This is made possible through the introduction of an appropriate definition of 2α in §1 enabling one to express shifting sentences as “typed properties” of the universe, in Boffa's sense. The disproof of AC is then completed in TT plus two extra ambiguity axioms. In §3, we show that this is in a sense the best possible result: that means that every single ambiguity axiom is consistent with TT plus AC, thus giving a positive solution to a conjecture of Specker [7, p. 119].

In this final chapter, we give a more detailed account of the role played by the Axiom of Choice (AC) in the theory of paradoxical decompositions. Ever since its discovery, the Banach-Tarski Paradox has caused many mathematicians to look critically at the Axiom of Choice. Indeed, as soon as the Hausdorff Paradox was discovered it was challenged because of its use of AC; E. Borel [21, p. 256] objected because the choice set was not explicitly defined. We shall discuss these criticisms in more detail later in this chapter, but first we deal with several technical points that are essential to understanding the connection between AC and the Banach-Tarski Paradox. Results of modern set theory can be used to show that AC is indeed necessary to obtain the Banach-Tarski Paradox, in the sense that the paradox is not a theorem of ZF alone. Before we can explain why this is so we need to introduce some notation and discuss some technical points of set theory. If T is a collection of sentences in the language of set theory, for example, T = ZF or T = ZF + AC, then Con (T) is the assertion, also a statement of set theory in fact, that T is consistent, that is, that a contradiction cannot be derived from T using the usual methods of proof. We take Con(ZF) as an underlying assumption in all that follows. Godel proved in 1938 that Con(ZF) implies (and so is equivalent to) Con(ZF + AC); thus AC does not contradict ZF (see [98, 99]).

The usefulness of measurable cardinals in set theory arises in good part from the fact that an ultraproduct of wellfounded structures by a countably complete ultrafilter is wellfounded. In the standard proof of the wellfoundedness of such an ultraproduct, one first shows, without any use of the axiom of choice, that the ultraproduct contains no infinite descending chains. One then completes the proof by noting that, assuming the axiom of choice, any partial ordering with no infinite descending chain is wellfounded. In fact, the axiom of dependent choices (a weakened form of the axiom of choice) suffices. It is therefore of interest to ask whether some use of the axiom of choice is needed in order to prove the wellfoundedness of such ultraproducts or whether, on the other hand, their wellfoundedness can be proved in ZF alone. In Theorem 1, we show that the axiom of choice is needed for the proof (assuming the consistency of a strong partition relation). Theorem 1 also contains some related consistency results concerning infinite exponent partition relations. We then use Theorem 1 to show how to change the cofinality of a cardinal κ satisfying certain partition relations to any regular cardinal less than κ, while introducing no new bounded subsets of κ. This generalizes a theorem of Prikry [5].

A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.