Elliott Mendelson. Some proofs of independence in axiomatic set theory. The journal of symbolic logic, vol. 21 (1956), pp. 291–303. - Elliott Mendelson. The independence of a weak axiom of choice. The journal of symbolic logic, pp. 350–366.

1958 ◽  
Vol 23 (1) ◽  
pp. 42-44
Author(s):  
Dana Scott
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Symbolic Logic ◽  
Axiomatic Set Theory ◽  
Weak Axiom

Note on two theorems of Mostowski

1945 ◽  
Vol 10 (1) ◽  
pp. 13-15 ◽  
Author(s):  
Raouf Doss
Keyword(s):  
Set Theory ◽  
Present Note ◽  
Axiom Of Choice ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory

The present note is conceived as a sequel to Mostowki's paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip (which will be quoted as “M.”).In that paper Mostowski considers a system of axioms , very close to the system of axiomatic set theory of Bernays, but where the axiom of choice is not supposed to hold. Mostowski proves that the well-ordering theorem (Wohlordnungssatz) cannot be derived from the system plus the principle of simple ordering (Ordnungsprinzip).

A system of axiomatic set theory—Part VI

1948 ◽  
Vol 13 (2) ◽  
pp. 65-79 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Axiom Of Choice ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory ◽  
Present Section

Comparability of classes. Till now we tried to get along without the axioms Vc and Vd. We found that this is possible in number theory and analysis as well as in general set theory, even keeping in the main to the usual way of procedure.For the considerations of the present section application of the axioms Vc, Vd is essential. Our axiomatic basis here consists of the axioms I—III, V*, Vc, and Vd. From V*, as we know, Va and Vb are derivable. We here take axiom V* in order to separate the arguments requiring the axiom of choice from the others. Instead of the two axioms V* and Vc, as was observed in Part II, V** may be taken as well.

A FINITE-TO-ONE MAP FROM THE PERMUTATIONS ON A SET

2016 ◽  
Vol 95 (2) ◽  
pp. 177-182 ◽  
Author(s):  
NATTAPON SONPANOW ◽  
PIMPEN VEJJAJIVA
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Symbolic Logic ◽  
The Axiom Of Choice ◽  
The Relationship ◽  
Infinite Set

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.

The independence of a weak axiom of choice

1956 ◽  
Vol 21 (4) ◽  
pp. 350-366 ◽  
Author(s):  
Elliott Mendelson
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Weak Form ◽  
Axiom Of Choice ◽  
Transfinite Induction ◽  
Exact Nature ◽  
Generalized Continuum ◽  
Power Set ◽  
The Axiom Of Choice ◽  
Weak Axiom

1. The purpose of this paper is to show that, if the axioms of a system G of set theory are consistent, then it is impossible to prove from them the following weak form of the axiom of choice: (H1) For every denumerable set x of disjoint two-element sets, there is a set y, called a choice set for x, which contains exactly one element in common with each element of x. Among the axioms of the system G, we take, with minor modifications, Axioms A, B, C of Gödel [6]. Clearly, the independence of H1 implies that of all stronger propositions, including the general axiom of choice and the generalized continuum hypothesis.The proof depends upon ideas of Fraenkel and Mostowski, and proceeds in the following manner. Let a be a denumerable set of objects Δ0, Δ1, Δ2, …, the exact nature of which will be specified later. Let μj = {Δ2j, Δ2j+1} for each j, c = {μ0, μ1, μ2, …}, and b = [the sum set of a]. By transfinite induction, construct the class Vc which is the closure of b under the power-set operation. For each j, it is possible to define a 1–1 mapping of Vc onto itself with the following properties. The mapping preserves the ε-relation, or, more precisely, .

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.)

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