Principle Of Dependent Choices Research Articles

On Hausdorff operators in ZF$\mathsf {ZF}$

AbstractA 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”.

Juhász’s topological generalization of Neumer’s theorem may fail in 𝖹𝖥

In set theory without the Axiom of Choice ( A C \mathsf {AC} ), we investigate the open problem of the deductive strength of Juhász’s topological generalization of Neumer’s Theorem from his paper On Neumer’s Theorem [Proc. Amer. Math. Soc. 54 (1976), 453–454]. Among other results, we show that Juhász’s Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of T 1 T_{1} spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen’s Countable Choice Principle, but does not imply either the full van Douwen’s Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász’s Theorem (for T 1 T_{1} spaces) implies each of the following principles: “ ℵ 1 \aleph _{1} is a regular cardinal”, “every infinite set is weakly Dedekind-infinite”, and “every infinite linearly ordered set is Dedekind-infinite”. We also establish that Juhász’s Theorem for T 2 T_{2} spaces is not provable in Z F \mathsf {ZF} . In contrast to the above results, we show that Neumer’s Theorem and Juhász’s Theorem for compact T 1 T_{1} spaces are both provable in Z F \mathsf {ZF} .

First Steps Towards a Formalization of Forcing

We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize a version of the principle of Dependent Choices and using it we prove the Rasiowa-Sikorski lemma on the existence of generic filters.Given a transitive set M, we define its generic extension M[G], the canonical names for elements of M, and finally show that if M satisfies the axiom of pairing, then M[G] also does. We also prove that M[G] is transitive.

Brezis-Browder Principle and Dependent Choice

The Brezis-Browder Ordering Principle is equivalent with the Principle of Dependent Choices; and as such, equivalent with Ekeland’s Variational Principle. Mathematics Subject Classification 2000: 49J53, 47J30.

Products of some special compact spaces and restricted forms of AC

AbstractWe establish the following results:1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent:(a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact.(b) The union of ∣α∣ many non-empty finite subsets of I is well orderable.2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory.3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity).4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

Well-foundedness in Realizability

However, the implication from 1) to 2) is equivalent to the principle of Dependent Choices (as the reader can show for himself) and cannot be proved in ZF. A relation < on X is called well-founded if any of these statements holds. Intuitionistically, the 3 statements are far from equivalent (hence, in formalizing an intuitionistic notion of well-foundedness, care is needed). Notion 1) is, intuitionistically, too weak to be of any use; whereas notion 2) is far too strong (by a well-known argument, as soon as an inhabited relation < satisfies 2), classical logic is forced on us). Hence we focus on notion 3), which is also usually taken as part of an axiomatization of intuitionistic set theory. The behaviour of well-founded induction for primitive recursive well-founded relations on the natural numbers in formal arithmetic, has been studied by many people. Classical results in the area are:

Compactness under constructive scrutiny

AbstractHow are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop‐style constructive mathematics as the basic system – that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by‐product, the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine‐Borel property for coverings by countably many possibly empty open balls. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Doughnuts, floating ordinals, square brackets, and ultraflitters

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

Dependent Choices and Weak Compactness

We work in set theory without the Axiom of Choice ZF. We prove that the Principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact and, in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF and the latter statement does not imply DC. Furthermore, DC does not imply that the closed unit ball of a reflexive space is weakly compact.

The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice

AbstractTwo Fraenkel‐Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2 and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.Mathematics Subject Classification: 03E25, 03E35, 04A25.

A Remark on Ascending Chain Conditions, the Countable Axiom of Choice and the Principle of Dependent Choices

AbstractIt is easy to prove in ZF− (= Zermelo‐Fraenkel set theory without the axioms of choice and foundation) that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well‐founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the (finite ascending) chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore (if ZF− is consistent). More precisely, we will prove that this statement is equivalent in ZF− to the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski.Mathematics Subject Classification: 06B05, 08A65, 08B20, 03E99.

On the Principle of Dependent Choices and Some Forms of Zorn's Lemma

AbstractThe main result of this paper is to prove that a generalization of the Principle of Dependent Choices, introduced by A. Levy [2; see also 1, Chapter 8], is equivalent to a form of Zorn's Lemma.

On the principle of dependent choices

Publisher Summary This chapter discusses the general axiom of choice (denoted by (Z)) is independent of (T), that is, cannot be proved from (T) and the usual axioms of set-theory. An independence-proof has sense only with respect to a well-defined formal system whose consistency is either proved or assumed as an hypothesis. Proof applies only to such systems of set-theory that remain self-consistent after the adjunction of the following axiom: there is a non-denumerable set of elements which are not sets.