Chapter II Axiomatic Foundations of Set Theory. The Axiom of Choice

This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

Set Theory with the Axiom of Choice

Classic Set Theory ◽

10.1201/9781315139432-9 ◽

2017 ◽

pp. 263-280

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Products of some special compact spaces and restricted forms of AC

10.2178/jsl/1278682212 ◽

2010 ◽

Vol 75 (3) ◽

pp. 996-1006 ◽

Cited By ~ 2

Author(s):

Kyriakos Keremedis ◽

Eleftherios Tachtsis

Keyword(s):

Set Theory ◽

Closed Subset ◽

Choice Function ◽

Ordinal Number ◽

Axiom Of Choice ◽

Finite Support ◽

Compact Spaces ◽

The Axiom Of Choice ◽

Infinite Set

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.

Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

Axioms ◽

10.3390/axioms7040086 ◽

2018 ◽

Vol 7 (4) ◽

pp. 86 ◽

Cited By ~ 1

Author(s):

Dmitri Shakhmatov ◽

Víctor Yañez

Keyword(s):

Topological Group ◽

Set Theory ◽

Axiom Of Choice ◽

Natural Numbers ◽

Compactness Property ◽

Free Ultrafilter ◽

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

A Theory of Mathematical Objects as a Prototype of Set Theory

Nagoya Mathematical Journal ◽

10.1017/s0027763000023680 ◽

1962 ◽

Vol 20 ◽

pp. 105-168 ◽

Cited By ~ 5

Author(s):

Katuzi Ono

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Simple System ◽

Object Theory ◽

Mathematical Objects ◽

Axiom Scheme ◽

The Axiom Of Choice ◽

Primitive Notion

The theory of mathematical objects, developed in this work, is a trial system intended to be a prototype of set theory. It concerns, with respect to the only one primitive notion “proto-membership”, with a field of mathematical objects which we shall hereafter simply call objects, it is a very simple system, because it assumes only one axiom scheme which is formally similar to the aussonderung axiom of set theory. We shall show that in our object theory we can construct a theory of sets which is stronger than the Zermelo set-theory [1] without the axiom of choice.

Krivine's classical realisability from a categorical perspective

Mathematical Structures in Computer Science ◽

10.1017/s0960129512000989 ◽

2013 ◽

Vol 23 (6) ◽

pp. 1234-1256 ◽

Cited By ~ 6

Author(s):

THOMAS STREICHER

Keyword(s):

Recent Work ◽

Set Theory ◽

Axiom Of Choice ◽

Second Order ◽

Order Logic ◽

Current Paper ◽

Categorical Approach ◽

The Axiom Of Choice ◽

Second Order Logic

In a sequence of papers (Krivine 2001; Krivine 2003; Krivine 2009), J.-L. Krivine introduced his notion of classical realisability for classical second-order logic and Zermelo–Fraenkel set theory. Moreover, in more recent work (Krivine 2008), he has considered forcing constructions on top of it with the ultimate aim of providing a realisability interpretation for the axiom of choice.The aim of the current paper is to show how Krivine's classical realisability can be understood as an instance of the categorical approach to realisability as started by Martin Hyland in Hyland (1982) and described in detail in van Oosten (2008). Moreover, we will give an intuitive explanation of the iteration of realisability as described in Krivine (2008).

Kurt Gödel. The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. (Lectures delivered at the Institute for Advanced Study 1938–1939; notes by George W. Brown.) Annals of Mathematics studies, no. 3. Lithoprinted. Princeton University Press, Princeton1940, 66 pp.

10.2307/2268607 ◽

1941 ◽

Vol 6 (3) ◽

pp. 112-114 ◽

Cited By ~ 1

Author(s):

Paul Bernays

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Axiom Of Choice ◽

Princeton University ◽

Advanced Study ◽

Kurt Gödel ◽

Generalized Continuum ◽

On generic extensions without the axiom of choice

10.2307/2273318 ◽

1983 ◽

Vol 48 (1) ◽

pp. 39-52 ◽

Cited By ~ 5

Author(s):

G. P. Monro

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Generic Extension ◽

Transitive Model ◽

The Axiom Of Choice ◽

Generic Extensions

AbstractLet ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a “generic extension”). If the axiom of choice holds in M it also holds in M[G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes.

On Arbitrary sets andZFC

Bulletin of Symbolic Logic ◽

10.2178/bsl/1309952318 ◽

2011 ◽

Vol 17 (3) ◽

pp. 361-393 ◽

Cited By ~ 8

Author(s):

José Ferreirós

Keyword(s):

Set Theory ◽

Axiom Of Choice ◽

Real Numbers ◽

The Real ◽

Formal Systems ◽

Infinite Sets ◽

The Axiom Of Choice ◽

The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

Relative consistency of an extension of Ackermann's set theory

10.1017/s0022481200051525 ◽

1976 ◽

Vol 41 (2) ◽

pp. 465-466

Author(s):

John Lake

Keyword(s):

Set Theory ◽

Relative Strength ◽

Axiom Of Choice ◽

Relative Consistency ◽

Free Variables ◽

Image Position ◽

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y1, …, yn, z is z1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we haveholding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).