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logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated
Projective_determinacy
Subfield of set theory
then projective determinacy holds; that is, every game whose winning condition is a projective set is determined. From projective determinacy it follows
Determinacy
Possible axiom for set theory
game is determined if it has a projective set as its winning set (see Projective determinacy). The axiom of determinacy implies that for every subspace
Axiom_of_determinacy
Mathematical system
projective determinacy[citation needed], that is a statement in the language of second-order arithmetic is provable in Z2 with projective determinacy if and
Second-order_arithmetic
Theorem in descriptive set theory
In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that
Borel_determinacy_theorem
Subfield of mathematical logic
Baire. This is related to the fact that ZFC proves Borel determinacy, but not projective determinacy. There are also generic extensions of L {\displaystyle
Descriptive_set_theory
(ordered set of) rational numbers QPD Quasi-projective determinacy quantifier ∀ or ∃ Quasi-projective determinacy All sets of reals in L(R) are determined
Glossary_of_set_theory
Prewellordering Projective set Property of Baire Uniformization (set theory) Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy
List_of_set_theory_topics
generalization Freiling's axiom of symmetry Axiom of determinacy Axiom of projective determinacy Martin's axiom Axiom of constructibility Rank-into-rank Kripke–Platek
List_of_axioms
Kind of large cardinal number
infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire
Woodin_cardinal
sets are determined. Projective determinacy Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American
Homogeneously_Suslin_set
Deductive system in set theory
theory of determinacy of pointclasses to cover the structure H ℵ 2 {\displaystyle H_{\aleph _{2}}} . Just as the axiom of projective determinacy yields a
Ω-logic
American mathematician (born 1948)
contributions to the theory of inner models and determinacy. With Donald A. Martin, he proved projective determinacy, assuming the existence of sufficiently large
John_R._Steel
American mathematician (born 1940)
analytic determinacy (from the existence of a measurable cardinal), Borel determinacy (from ZFC alone), the proof (with John R. Steel) of projective determinacy
Donald_A._Martin
Topics referred to by the same term
Photodynamic Therapy Point-defence, a category of weapons Axiom of projective determinacy, in mathematical logic Pumpe Düse, a Volkswagen Group name for Unit
PD
and Steel's proof of projective determinacy. Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American
Homogeneous_tree
measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable
Universally_measurable_set
Set theory axiom extension
clause by itself is referred to as ordinal determinacy. Axiom of projective determinacy Axiom of real determinacy Suslin's problem Topological game Larson
AD+
Prewellordering Projective set Property of Baire Uniformization (set theory) Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy
List of mathematical logic topics
List_of_mathematical_logic_topics
Existence of 0# Singular cardinals hypothesis Projective determinacy (and even the full axiom of determinacy if the axiom of choice is not assumed) There
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Axiom of set theory
of projective determinacy Topological game Ikegami, Daisuke; de Kloet, David; Löwe, Benedikt (2012-11-01). "The axiom of real Blackwell determinacy". Archive
Axiom_of_real_determinacy
Difference of an open set by a meager set
Therefore, it follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set (in a Polish space) has
Property_of_Baire
Infinite game in descriptive set theory whose payoff set is a lightface analytic set
Analytic set Analytical hierarchy Borel determinacy theorem Descriptive set theory Projective determinacy Zero sharp Jech, Thomas (2003). Set Theory
Lightface_analytic_game
Theories in mathematical logic
constructibility (V=L) Proper forcing axiom Analytic determinacy, projective determinacy, Axiom of determinacy Many large cardinal axioms Mathematics portal
List_of_first-order_theories
Smallest transitive inner model of ZF containing all the ordinals and all the reals
satisfies ZF + AD+. (In particular, it satisfies the axiom of determinacy.) Every projective set of reals – and therefore every analytic set and every Borel
L(R)
Branch of mathematics that studies sets
is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom
Set_theory
German mathematician and logician
Berkeley. In 2001, he completed his PhD entitled Blackwell Determinacy about determinacy under supervision of Donald A. Martin and Ronald Björn Jensen
Benedikt_Löwe
American mathematician (born 1955)
has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, bears his name. In
W._Hugh_Woodin
sets, Σ1n sets, or Π1n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma
Wadge_hierarchy
American mathematician
consequences of the axiom of determinacy. In particular he is known for having calculated the values of all the projective ordinals (the suprema of the
Steve_Jackson_(mathematician)
Concept in mathematical logic and set theory
formulas that can be used to define them; it is the lightface version of the projective hierarchy. The notation Σ 0 1 = Π 0 1 = Δ 0 1 {\displaystyle \Sigma _{0}^{1}=\Pi
Analytical_hierarchy
Finite ordered list of elements
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Tuple
descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player integer
Moschovakis_coding_lemma
Descriptive set theory concept
analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses
Pointclass
Wadge hierarchy, they lie above the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class of inductive sets has the
Inductive_set
Mathematician (1845–1918)
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Georg_Cantor
Hungarian and American mathematician and physicist (1903–1957)
result in projective geometry to the continuous dimensional case. This coordinatization theorem stimulated considerable work in abstract projective geometry
John_von_Neumann
Axiom of set theory
Every free abelian group is projective. Baer's criterion: Every divisible abelian group is injective. Every set is a projective object in the category of
Axiom_of_choice
Alternative to the standard Zermelo–Fraenkel set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
List of alternative set theories
List_of_alternative_set_theories
Problem in set theory
cardinals and all regular successor cardinals—it implies that the axiom of determinacy holds in L(R) and is believed to imply the existence of an inner model
Suslin's_problem
Mathematical concept
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Equivalence_class
German mathematician (1831–1916)
Mathematics Archive, University of St Andrews Works by Richard Dedekind at Project Gutenberg Works by or about Richard Dedekind at the Internet Archive Dedekind
Richard_Dedekind
Axiom of Zermelo-Fraenkel set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_infinity
Technique invented by Paul Cohen for proving consistency and independence results
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Forcing_(mathematics)
American mathematician
Paris–Harrington theorem along with Jeff Paris, showing that if the axiom of determinacy holds for all analytic sets then x# exists for all reals x, and proving
Leo_Harrington
Use of braces for specifying sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Set-builder_notation
uniformized in some transitive inner model of V in which the axiom of determinacy holds.) Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North
Uniformization_(set_theory)
Hyperarithmetical set Analytical set Analytic set, Coanalytic set Suslin set Projective set Inhabited set Multiset List of set identities and relations – Equalities
List_of_types_of_sets
Abstract strategy board game
published a proof that the determinacy of Hex is equivalent to the two-dimensional Brouwer fixed-point theorem, and that the determinacy of higher-dimensional
Hex_(board_game)
Finite sets whose elements are all hereditarily finite sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Hereditarily_finite_set
Israeli mathematician
Angeles. He has made major contributions to the theory of inner models, determinacy and forcing. Neeman was born in 1972 in Safed, Israel. After studying
Itay_Neeman
Pair of logical equivalences
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
De_Morgan's_laws
Sets whose elements have degrees of membership
Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, Project VOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences
Fuzzy_set
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Nested_set_collection
(model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical logic)
List_of_theorems
Axiom of set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_regularity
Term in set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Almost
Facts provided or learned about something or someone
in 2020. Over the next five years up to 2025, global data creation is projected to grow to more than 180 zettabytes. Records are specialized forms of
Information
Elements in exactly one of two sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Symmetric_difference
Size of a set in mathematics
multiverse of set theories, but no "absolute" or "true" model. The Axiom of Determinacy (AD) asserts that certain kinds of mathematical games on the natural
Cardinality
Logical paradox in decision-making theory
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Paradox_of_tolerance
Set with exactly one element
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Singleton_(mathematics)
Concept in mathematics
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_countable_choice
Possible axiom of set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_limitation_of_size
Paradox in set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Burali-Forti_paradox
Concept in axiomatic set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_union
American mathematician (1934–2007)
17, 2007. Retrieved 2007-10-31. Solomon Feferman, The Gödel Editorial Project: A synopsis [1] p. 11. Pearce, Jeremy (2007-04-02). "Paul J. Cohen, Mathematics
Paul_Cohen
In mathematics, operation on sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Disjoint_union
Swiss mathematician (1888–1977)
Commons has media related to Paul Bernays (mathematician). Hilbert Bernays Project O'Connor, John J.; Robertson, Edmund F., "Paul Bernays", MacTutor History
Paul_Bernays
Infinite set not splittable into infinite sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Amorphous_set
System of mathematical set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Zermelo_set_theory
Concept in axiomatic set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_schema_of_specification
German-Israeli mathematician and Zionist (1891–1965)
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Abraham_Fraenkel
Polish mathematician (1887–1972)
together with another of his students, Jan Mycielski, proposed the axiom of determinacy. Steinhaus was also an early contributor to, and co-founder of, probability
Hugo_Steinhaus
Concept in game theory
among a group of players who have collaborated. For example, in a team project where each member contributed differently, the Shapley value provides a
Shapley_value
Greek-French philosopher (1922–1997)
dissertation on mathematical logic to Poirier, but by 1948 he had abandoned the project. The working title of his unfinished thesis was Introduction à la logique
Cornelius_Castoriadis
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Gödel_logic
Czech mathematician
1090/S0273-0979-1980-14818-1. Home page Archived 2012-05-04 at the Wayback Machine, with a copy at Penn state. Thomas Jech at the Mathematics Genealogy Project
Thomas_Jech
Solution concept of a non-cooperative game
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Nash_equilibrium
Concept in axiomatic set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_power_set
Family of subsets representing "large" sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Filter_on_a_set
Paper-and-pencil game for two players
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Tic-tac-toe
American economist and Nobel Laureate (born 1943)
monetary-policy instruments and rules on output stability and price determinacy. help make the theory of rational expectations statistically operational
Thomas_J._Sargent
Search algorithm
Alan (2004) [1962]. "A Chess Playing Program". Artificial Intelligence Project. RLE and MIT Computation Center. Memo 41. Retrieved 2006-07-01. Marsland
Alpha–beta_pruning
3-volume treatise on mathematics, 1910–1913
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Principia_Mathematica
Generalization of "n-th" to infinite cases
theory without the Axiom of Choice, such as those satisfying the Axiom of determinacy, the club filter on ω 1 {\displaystyle \omega _{1}} can be an ultrafilter
Ordinal_number
and Brigade Commander Mykola from the Shkval Battalion praised their determinacy and qualitative combat performance. Women in the Ukrainian military Middle
Women drone operators in the Ukrainian military
Women_drone_operators_in_the_Ukrainian_military
Mathematical logic hierarchy
Logic, Springer-Verlag (1978). ISBN 3-540-07904-1. D. Martin, Borel Determinacy, Annals of Mathematics vol. 102, pp.363--371 (1975) Kechris, Alexander
Borel_hierarchy
Decision rule used for minimizing the possible loss for a worst-case scenario
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Minimax
Situation where total gains match total losses
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Zero-sum_game
intimate with both his sister and his mother, and used that to question his determinacy, which led to Yuu not able to fare better against him this time. Back
List_of_Brain_Powerd_episodes
Axiom used in set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_extensionality
Game theory case weighing own/others' sacrifice
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Volunteer's_dilemma
Model of conflict for two players in game theory
behavior without help from computers. The term "schedule chicken" is used in project management and software development circles. The condition occurs when
Chicken_(game)
Combinatorial game theory theorem
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Sprague–Grundy_theorem
Technique of using algorithms to create music
Western counterpoint, for example, can often be reduced to algorithmic determinacy. The term can be used to describe music-generating techniques that run
Algorithmic_composition
Paradox in set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Russell's_paradox
Concept in axiomatic set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Axiom_of_pairing
Israeli psychologist (1937–1996)
smarter than you, the smarter you were." Michael Lewis's book The Undoing Project: A Friendship That Changed Our Minds, released in 2016, is about Tversky's
Amos_Tversky
Mathematical set formed from two given sets
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity
Cartesian_product
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