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.

scholarly journals A Stronger System of Object Theory as a Prototype of Set Theory

1963 ◽  
Vol 22 ◽  
pp. 119-167 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Object Theory ◽  
Mathematical Objects ◽  
The Axiom Of Choice ◽  
New System

We have introduced in our former work [1] a theory of mathematical objects which can be regarded as a prototype of set theory. We have been successful to imbed the Zermelo set-theory [3] without the axiom of choice in our system. However, it seems impossible to imbed the Fraenkel set-theory [4] in our system even without the axiom of choice. In this work, we introduce another system of object theory in which we can imbed the Fraenkel set-theory without the axiom of choice. We shall denote our former system by OZ (object theory in the manner of the Zermelo set-theory) and the new system we are going to introduce in this work by OF (object theory in the manner of the Fraenkel set-theory). We shall also denote the Zermelo set-theory without the axiom of choice by SZ and the Fraenkel set-theory without the axiom of choice by SF.

scholarly journals On a Theory Objects Based on a Single Axiom Scheme

1966 ◽  
Vol 26 ◽  
pp. 13-30
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Axiom Of Choice ◽  
Fundamental Theory ◽  
Axiom Scheme ◽  
Single Axiom ◽  
The Axiom Of Choice ◽  
Formal Theories

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

scholarly journals High-order metaphysics as high-order abstractions and choice in set theory

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
High Order ◽  
Precise Definition ◽  
The Other ◽  
Axiom Scheme ◽  
The One ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Comprehension Axiom

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc.

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.

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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