Andrzej Mostowski. Axiom of choice for finite sets. Fundamenta mathematicae, vol. 33 (1945), pp. 137–168.

1948 ◽  
Vol 13 (1) ◽  
pp. 46-48
Author(s):  
John G. Kemeny
Keyword(s):  
Axiom Of Choice ◽  
Finite Sets

scholarly journals On a cardinal equation in set theory

1972 ◽  
Vol 6 (3) ◽  
pp. 447-457 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Linear Ordering ◽  
Finite Sets ◽  
Intrinsic Interest ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Finiteness Criterion

We work in a Zermelo-Fraenkel set theory without the Axiom of Choice. In the appendix to his paper “Sur les ensembles finis”, Tarski proposed a finiteness criterion that we have called “C-finiteness”: a nonempty set is called “C-finite” if it cannot be partitioned into two blocks, each block being equivalent to the whole set. Despite the fact that this criterion can be shown to possess several features that are undesirable in a finiteness criterion, it has a fair amount of intrinsic interest. In Section 1 of this paper we look at a certain class of C-finite sets; in Section 2 we derive a few consequences from the negation of C-finiteness; and in Section 3 we show that not every C-infinite set necessarily possesses a linear ordering. Any unexplained notation is given in my paper, “Some definitions of finiteness”, Bull. Austral. Math. Soc. 5 (1971).

scholarly journals How to have more things by forgetting how to count them

2020 ◽  
Vol 476 (2239) ◽  
pp. 20190782
Author(s):  
Asaf Karagila ◽  
Philipp Schlicht
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Ground Model ◽  
Finite Sets ◽  
Real Numbers ◽  
Extremally Disconnected ◽  
Finite Set ◽  
Combinatorial Condition ◽  
The Axiom Of Choice

Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

Independence results concerning Dedekindfinite sets

1975 ◽  
Vol 19 (1) ◽  
pp. 35-46 ◽  
Author(s):  
G. P. Monro
Keyword(s):  
Axiom Of Choice ◽  
Finite Sets ◽  
Finite Set ◽  
Independence Results ◽  
The Axiom Of Choice

A Dedekind-finite set is one not equinumerous with any of its proper subsets; it is well known that the axiom of choice implies that all such sets are finite. In this paper we show that in the absence of the axiom of choice it is possible to construct Dedekind-finite sets which are large, in the sense that they can be mapped onto large ordinals; we extend the result to proper classes. It is also shown that the axiom of choice for countable sets is not implied by the assumption that all Dedekind-finite sets are finite.

Almost combinatorial Skolem functions

1970 ◽  
Vol 35 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Finite Sets ◽  
First Order ◽  
Universal Sentence ◽  
Skolem Functions

Let be a version of class set theory admitting urelemente, and with AC (= axiom of choice) replaced by AC0 (= axiom of choice for sets of finite sets), ω = nonnegative integers, and Δ = Dedekind cardinals. Let be an arbitrarily quantified positive first order sentence in functors for + and ·. Let ƒ0, … , ƒκ - 1 be function variables and the universal sentence obtained from by replacing existential quantifiers by the ƒ1 as Skolem functions.

Ramsey's theorem in the hierarchy of choice principles

1977 ◽  
Vol 42 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Andreas Blass
Keyword(s):  
Axiom Of Choice ◽  
Extension Principle ◽  
Infinite Length ◽  
Finite Sets ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Dependent Choice ◽  
Weak Axioms Of Choice

Ramsey's theorem [5] asserts that every infinite set X has the following partition property (RP): For every partition of the set [X]2 of two-element subsets of X into two pieces, there is an infinite subset Y of X such that [Y]2 is included in one of the pieces. Ramsey explicitly indicated that his proof of this theorem used the axiom of choice. Kleinberg [3] showed that every proof of Ramsey's theorem must use the axiom of choice, although rather weak forms of this axiom suffice. J. Dawson has raised the question of the position of Ramsey's theorem in the hierarchy of weak axioms of choice.In this paper, we prove or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book [2]. Our results, along with some known facts which we include for completeness, may be summarized as follows (the notation being as in [2]):A. The following principles do not (even jointly) imply Ramsey's theorem, nor does Ramsey's theorem imply any of them:the Boolean prime ideal theorem,the selection principle,the order extension principle,the ordering principle,choice from wellordered sets (ACW),choice from finite sets,choice from pairs (C2).B. Each of the following principles implies Ramsey's theorem, but none of them follows from Ramsey's theorem:the axiom of choice,wellordered choice (∀kACk),dependent choice of any infinite length k (DCk),countable choice (ACN0),nonexistence of infinite Dedekind-finite sets (WN0).

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