scholarly journals REALIZING REALIZABILITY RESULTS WITH CLASSICAL CONSTRUCTIONS

2019 ◽  
Vol 25 (4) ◽  
pp. 429-445
Author(s):  
ASAF KARAGILA
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
New Method ◽  
The Axiom Of Choice ◽  
Models Of Set Theory

AbstractJ. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.

Zermelo and Set Theory

2004 ◽  
Vol 10 (4) ◽  
pp. 487-553 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
20Th Century ◽  
Set Theory ◽  
Historical Context ◽  
Subsequent Development ◽  
Axiom Of Choice ◽  
Mathematical Work ◽  
Infinitary Logic ◽  
Bernstein Theorem ◽  
The Axiom Of Choice ◽  
Models Of Set Theory

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

Cardinal collapsing and ordinal definability

1978 ◽  
Vol 43 (4) ◽  
pp. 635-642 ◽  
Author(s):  
Petr Štěpánek
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Complete Boolean Algebra ◽  
Ground Model ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
Generic Extensions ◽  
Models Of Set Theory

We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.Let be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if G is an -generic ultrafilter on a rigid complete Boolean algebra C, then every set in [G] is definable from parameters in . Hence if B is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in , it suffices to construct a rigid complete Boolean algebra C, in which B is completely embedded. If G is as above, then [G] satisfies “every set is -definable” and the inner model [G ∩ B] contains the collapsing mapping determined by B. To complete the result, it is necessary to give some conditions under which every cardinal from [G ∩ B] remains a cardinal in [G].The absolutness is granted for every cardinal at least as large as the saturation of C. To keep the upper cardinals absolute, it often suffices to construct C with the same saturation as B. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.

scholarly journals A remark on elementary abelian groups

1977 ◽  
Vol 16 (2) ◽  
pp. 213-217 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Set Theory ◽  
Abelian Groups ◽  
Axiom Of Choice ◽  
Infinite Cardinality ◽  
The Axiom Of Choice ◽  
Finite Exponent ◽  
Models Of Set Theory

Dr M.F. Newman has asked whether in the absence of the Axiom of Choice it is possible to have two non-isomorphic elementary abelian groups of the same (finite) exponent and of the same (infinite) cardinality. By means of an example, I show that this is in fact possible, if the exponent is at least five; I do not know the answer in the remaining two cases. The example given requires the construction of a Fraenkel-Mostowski model of set theory, and for this purpose I draw upon the terminology, constructions, and results contained in the first two sections of a previous paper, “The construction of groups in models of set theory that fail the Axiom of Choice” (Bull. Austral. Math. Soc. 14 (1976), 199–232), with which I assume familiarity.

Ultrapowers without the axiom of choice

1988 ◽  
Vol 53 (4) ◽  
pp. 1208-1219 ◽  
Author(s):  
Mitchell Spector
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Fundamental Theorem ◽  
Axiom Of Choice ◽  
Partition Relations ◽  
Omitting Types ◽  
The Axiom Of Choice ◽  
Almost All ◽  
Models Of Set Theory ◽  
Theory And Model

AbstractA new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

Models of set theory with more real numbers than ordinals

1974 ◽  
Vol 39 (3) ◽  
pp. 579-583 ◽  
Author(s):  
Paul E. Cohen
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Infinite Cardinal ◽  
Real Numbers ◽  
Transitive Model ◽  
Lower Case ◽  
The Axiom Of Choice ◽  
Dependent Choice ◽  
Models Of Set Theory

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

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.

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