Independence results for class forms of the axiom of choice

1978 ◽  
Vol 43 (4) ◽  
pp. 673-684 ◽  
Author(s):  
Paul E. Howard ◽  
Arthur L. Rubin ◽  
Jean E. Rubin
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Von Neumann ◽  
Independence Results ◽  
The Axiom Of Choice

AbstractLet NBG be von Neumann-Bemays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.

scholarly journals Foundations of Transmathematics

2019 ◽  
Author(s):  
James A.D.W. Anderson
Keyword(s):  
Set Theory ◽  
Paraconsistent Logic ◽  
Axiom Of Choice ◽  
Von Neumann ◽  
Naive Set Theory ◽  
The Axiom Of Choice ◽  
Do So

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics. Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic. In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known. As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

scholarly journals A system of axiomatic set theory. Part IV. General set theory

1942 ◽  
Vol 7 (4) ◽  
pp. 133-145 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Point Of View ◽  
The Other ◽  
Von Neumann ◽  
Neumann System ◽  
Other Hand ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

The Axiom of Choice Machine

2018 ◽  
Author(s):  
Alexander R. Pruss
Keyword(s):  
Set Theory ◽  
Laws Of Nature ◽  
Axiom Of Choice ◽  
Dutch Book ◽  
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.

Products of some special compact spaces and restricted forms of AC

2010 ◽  
Vol 75 (3) ◽  
pp. 996-1006 ◽  
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.

scholarly journals Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 86 ◽  
Author(s):  
Dmitri Shakhmatov ◽  
Víctor Yañez
Keyword(s):  
Topological Group ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Natural Numbers ◽  
Compactness Property ◽  
Free Ultrafilter ◽  
The Axiom Of Choice

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.

scholarly journals A Theory of Mathematical Objects as a Prototype of Set Theory

1962 ◽  
Vol 20 ◽  
pp. 105-168 ◽  
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

2013 ◽  
Vol 23 (6) ◽  
pp. 1234-1256 ◽  
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).

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