Kurt Gödel. The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proceedings of the National Academy of Sciences, vol. 24 (1938), pp. 556–557.

1940 ◽  
Vol 5 (3) ◽  
pp. 116-117
Author(s):  
Paul Bernays
Keyword(s):  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
National Academy Of Sciences ◽  
Kurt Gödel ◽  
Generalized Continuum ◽  
The Axiom Of Choice ◽  
Academy Of Sciences

On the inadequacy of inner models

1972 ◽  
Vol 37 (3) ◽  
pp. 569-571
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Precise Statement ◽  
Relative Consistency ◽  
Inner Models ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The method of inner models, used by Gödel to prove the (relative) consistency of the axiom of choice and the generalized continuum hypothesis [2], cannot be used to prove the (relative) consistency of any statement which contradicts the axiom of constructibility (V = L). A more precise statement of this well-known fact is:(*)For any formula θ(x) of the language of ZF, there is an axiom α of the theory ZF + V ≠ L such that the relativization α(θ) is not a theorem of ZF.On p. 108 of [1], Cohen gives a proof of (*) in ZF assuming the existence of a standard model of ZF, and he indicates that this assumption can be avoided. However, (*) is not a theorem of ZF (unless ZF is inconsistent), because (*) trivially implies the consistency of ZF. What assumptions are needed to prove (*)? We know that the existence of a standard model implies (*) which, in turn, implies the consistency of ZF. Is either implication reversible?From our main result, it will follow that, if the converse of the first implication is provable in ZF, then ZF has no standard model, and if the converse of the second implication is provable in ZF, then so is the inconsistency of ZF. Thus, it is quite improbable that either converse is provable in ZF.

Postulates for the calculus of binary relations

1940 ◽  
Vol 5 (3) ◽  
pp. 85-97 ◽  
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Binary Relations ◽  
Prove Theorem ◽  
Alfred Tarski ◽  
Cardinal Numbers ◽  
System A ◽  
Mathematical System ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The theory of relations was discussed, along with the theory of classes, by Peirce and Schröder. While the calculus of classes has subsequently been presented (by Couturat, and by Huntington) as an abstract mathematical system, no similar formulation has been published of the calculus of relations. Professor Alfred Tarski has proposed to me, the problem of developing a similar formulation for the theory of relations. In this paper I present a set of postulates for the calculus of relations.The postulates have been chosen so as to enable us to prove Theorem A, below, which asserts a kind of completeness of the system. A set of postulates is called semi-categorical, if every two realizations which involve the same number of elements are isomorphic. From Theorem A one can easily prove that my postulates are semi-categorical, if one assumes the following theorem from Mengenlehre: if α and β are cardinal numbers such that , then α=β. This theorem has not been proved, however, except on the assumption of the axiom of choice and the generalized continuum hypothesis, and is not assumed in the present paper.It should be pointed out, that the sort of completeness I have chosen is not the only sort, or even necessarily the most interesting sort, of completeness which a set of postulates for the calculus of relations could possess. For example, the question arises, whether it would be possible to find a set of postulates from which all true equations (involving free variables and the operations used by Schröder) could be derived, and which was the weakest possible such set. The present work should therefore be regarded only as a first step in the investigation of the axiomatization of the calculus of relations.

scholarly journals Cardinality of the sets of all bijections, injections and surjections

2020 ◽  
Vol 11 ◽  
pp. 143-151
Author(s):  
Marcin Zieliński
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections is considered in detail and more explicit an expression is obtained when the Generalized Continuum Hypothesis is assumed.

The constructible universe

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Constructible Sets ◽  
Kurt Gödel ◽  
The Axiom Of Choice ◽  
The Universe ◽  
The Continuum

the ‘universe’ of constructible sets was introduced by Kurt Gödel in order to prove the consistency of the axiom of choice (AC) and the continuum hypothesis (CH) with the basic (ZF) axioms of set theory. The hypothesis that all sets are constructible is the axiom of constructibility (V = L). Gödel showed that if ZF is consistent, then ZF + V = L is consistent, and that AC and CH are provable in ZF + V = L.

Sign in / Sign up

Export Citation Format

Share Document