scholarly journals The axiom of determinacy implies dependent choice in mice

2019 ◽  
Vol 65 (3) ◽  
pp. 370-375
Author(s):  
Sandra Müller
Keyword(s):  
Axiom Of Determinacy ◽  
Dependent Choice

Some consequences of an infinite-exponent partition relation

1977 ◽  
Vol 42 (4) ◽  
pp. 523-526 ◽  
Author(s):  
J. M. Henle
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Partition Relation ◽  
Basic Theorem ◽  
Axiom Of Determinacy ◽  
Ramsey’S Theorem ◽  
Partition Relations ◽  
Dependent Choice

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

AD and the supercompactness of ℵ1

1981 ◽  
Vol 46 (4) ◽  
pp. 822-842 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Axiom Of Choice ◽  
Axiom Of Determinacy ◽  
The Subject ◽  
The Axiom Of Choice ◽  
Dependent Choice

Since the discovery of forcing in the early sixties, it has been clear that many natural and interesting mathematical questions are not decidable from the classical axioms of set theory, ZFC. Therefore some mathematicians have been studying the consequences of stronger set theoretic assumptions. Two new types of axioms that have been the subject of much research are large cardinal axioms and axioms asserting the determinacy of definable games. The two appear at first glance to be unrelated; one of the most surprising discoveries of recent research is that this is not the case.In this paper we will be assuming the axiom of determinacy (AD) plus the axiom of dependent choice (DC). AD is false, since it contradicts the axiom of choice. However every set in L[R] is ordinal definable from a real. Our axiom that definable games are determined implies that every game in L[R] is determined (in V), and since a strategy is a real, it is determined in L[R]. That is, L[R] ⊨ AD. The axiom of choice implies L[R] ⊨ DC. So by embedding ourselves in L[R], we can assume AD + DC and begin proving theorems. These theorems true in L[R] imply corresponding theorems in V, by e.g. changing “every set” to “every set in L[R]”. For more information on AD as an axiom, and on some of the points touched on here, the reader should consult [14], particularly §§7D and 8I. In this paper L[R] will no longer even be mentioned. We just assume AD for the rest of the paper.

Relativized realizability in intuitionistic arithmetic of all finite types

1978 ◽  
Vol 43 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Extension Theorem ◽  
General Notion ◽  
Conservative Extension ◽  
First Order ◽  
New Information ◽  
Reflection Theorem ◽  
Order Arithmetic ◽  
Special Case ◽  
Dependent Choice ◽  
Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

MODIFIED BAR RECURSION AND CLASSICAL DEPENDENT CHOICE

2005 ◽  
pp. 89-107
Author(s):  
ULRICH BERGER ◽  
PAULO OLIVA
Keyword(s):  
Dependent Choice

The axiom of determinacy, strong partition properties, and nonsingular measures

2010 ◽  
pp. 333-354 ◽  
Author(s):  
Alexander S. Kechris ◽  
Eugene M. Kleinberg ◽  
Yiannis N. Moschovakis ◽  
W. Hugh Woodin
Keyword(s):  
Axiom Of Determinacy

scholarly journals Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

2020 ◽  
Vol 20 (03) ◽  
pp. 2050015
Author(s):  
Raphaël Carroy ◽  
Andrea Medini ◽  
Sandra Müller
Keyword(s):  
Homogeneous Space ◽  
General Result ◽  
Homogeneous Spaces ◽  
Locally Compact ◽  
Compact Spaces ◽  
Axiom Of Determinacy ◽  
Locally Compact Spaces

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

scholarly journals THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

Modified bar recursion and classical dependent choice

2017 ◽  
pp. 89-107 ◽  
Author(s):  
Ulrich Berger ◽  
Paulo Oliva
Keyword(s):  
Dependent Choice

Control of ribosome traffic by position-dependent choice of synonymous codons

2013 ◽  
Vol 10 (5) ◽  
pp. 056011 ◽  
Author(s):  
Namiko Mitarai ◽  
Steen Pedersen
Keyword(s):  
Synonymous Codons ◽  
Dependent Choice
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