Inconsistency of the Axiom of Choice with the positive theory

2000 ◽  
Vol 65 (4) ◽  
pp. 1911-1916
Author(s):  
Olivier Esser
Keyword(s):  
Axiom Of Choice ◽  
Positive Theory ◽  
Russell’S Paradox ◽  
Axiom Scheme ◽  
Russell's Paradox ◽  
The Axiom Of Choice

AbstractThe 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 .

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 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.

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