The Axiom of Choice

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...) algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. (shrink)

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without "too much" negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory $GPK^+ \infty$.

In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and motivations for, Russell’s (...) method of analysis, which are intended to shed light on his view about the status of mathematical axioms. I describe the position Russell develops in consequence as “immanent logicism,” in contrast to what Irving (1989) describes as “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament, and to propose a method for discovering structural relationships of dependence within mathematical theories. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that beefing up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.

A generalization of the axioms of choice says that all the Skolem functions of a true first‐order sentence exist. This generalization can be implemented on the first‐order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first‐order axiomatic set theory , it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what (...) the truth of S means in a combinational sense, in any model of FAST there are sentences which are set‐theoretical “true” but false in the normal sense of the word. This shows that the assumptions on which the axiom of choice rests cannot be fully implemented in FAST. The axiom choice is not a set‐theoretical principle. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom of Dependent (...) Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

Consider the following well-known result from the theory of normed linear spaces ([2], p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this (...) we derive various corollaries, for example: the conjunction of the Boolean prime ideal theorem and the Krein-Milman theorem implies the axiom of choice, and the Krein-Milman theorem is not derivable from the Boolean prime ideal theorem. (shrink)

Provability, Computability and Reflection.

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted to it, and it has long played a prominently role in discussions on the foundations of (...) mathematics. (shrink)

The principle of set theory known as the Axiom of Choice has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years ago” (Fraenkel, Bar-Hillel & Levy 1973, §II.4). The fulsomeness of this description might lead those unfamiliar with the axiom to expect it to be as startling as, say, the Principle of the Constancy of (...) the Velocity of Light or the Heisenberg Uncertainty Principle. But in fact the Axiom of Choice as it is usually stated appears humdrum, even self-evident. For it amounts to nothing more than the claim that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set — a transversal or choice set — containing exactly one element from each member of the given collection. Nevertheless, this seemingly innocuous principle has far-reaching mathematical consequences — many indispensable, some startling — and has come to figure prominently in discussions on the foundations of mathematics. It (or its equivalents) have been employed in countless mathematical papers, and a number of monographs have been exclusively devoted to it. (shrink)

. Dzik [2] gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems, here called Dzik systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional . A partial solution (...) is provided. (shrink)

We present a cut-elimination proof for simple type theory with an axiom of choice formulated in the language with an epsilon-symbol. The proof is modeled after Takahashi's proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classical logic with an axiom of choice.

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation.

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.