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Subfield of set theory
Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy
Determinacy
projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated
Projective_determinacy
Possible axiom for set theory
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962
Axiom_of_determinacy
Kinematic determinacy is a term used in structural mechanics to describe a structure where material compatibility conditions alone can be used to calculate
Kinematic_determinacy
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
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
Philosophical terms
article on a computer. But in an examination of the possibility of the determinacy of any possible concept, differences like this are supremely relevant
Indeterminacy_(philosophy)
Axiom of set theory
In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following: Axiom—Consider infinite two-person
Axiom_of_real_determinacy
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)
analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ {\displaystyle \mu } satisfies
Carleman's_condition
When a structure's static equilibrium equations have no unique solution
necessary for stability. Christian Otto Mohr Flexibility method Kinematic determinacy Overconstrained mechanism Structural engineering Matheson, James Adam
Statically_indeterminate
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
Facts provided or learned about something or someone
Information is an abstract concept that refers to something which has the power to inform. At the most fundamental level, it pertains to the interpretation
Information
Mathematical system
projective determinacy are hard to find. ZFC + {there are n Woodin cardinals: n is a natural number} is conservative over Z2 with projective determinacy[citation
Second-order_arithmetic
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
Russian mathematician and sociologist
to sociological analysis. He is best known for creating and defining determinacy analysis and the theory of rules. Sergey Chesnokov was born on 29 June
Sergey_Chesnokov
real determinacy Von Neumann–Bernays–Gödel axioms Continuum hypothesis and its generalization Freiling's axiom of symmetry Axiom of determinacy Axiom
List_of_axioms
Building or transport method that can change its shape at operator's will
state. In other words, a structure that has both statical determinacy and kinematic determinacy is optimal for actuation. Active-control technology is applied
Active_structure
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
Theory of equilibrium between supply and demand
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting
General_equilibrium_theory
American mathematician
involved the descriptive set-theoretic consequences of the axiom of determinacy. In particular he is known for having calculated the values of all the
Steve_Jackson_(mathematician)
Set theory axiom extension
theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DCR (the
AD+
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
Infinite Cardinal number
equinumerous with W α ( A ) {\displaystyle W_{\alpha }(A)} .) Borel determinacy is implied by the existence of all beths of countable index. Uncountable
Beth_number
Trying to map moments to a measure that generates them
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ {\displaystyle \mu } to the sequence of moments
Moment_problem
Topics referred to by the same term
Determined: A Science of Life Without Free Will, 2023 book by Robert Sapolsky Determinacy, a subfield of game theory and set theory This disambiguation page lists
Determined
Infinite game in descriptive set theory whose payoff set is a lightface analytic set
payoff set is a lightface analytic (Σ11) subset of Baire space. The determinacy of all lightface analytic games is equivalent, over ZFC, to the existence
Lightface_analytic_game
Structural engineering theory is the application of physics and mathematics to analyze and design structures to ensure they can withstand loads. Structural
Structural_engineering_theory
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
Branch of mathematical logic
recursion, ∆0 2 determinacy, and the ∆1 1 Ramsey theorem are all equivalent to each other. Over RCA0, Σ1 1 monotonic induction, Σ0 2 determinacy, and the Σ1
Reverse_mathematics
Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy Axiom of real determinacy Empty set Forcing (mathematics)
List_of_set_theory_topics
American mathematician (born 1940)
notable works are the proofs of analytic determinacy (from the existence of a measurable cardinal), Borel determinacy (from ZFC alone), the proof (with John
Donald_A._Martin
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
of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter. Let D {\displaystyle D} be the
Martin_measure
Broadest definition of sizes in integer-dimensional spaces
Lebesgue measure 0. If the axiom of determinacy holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with the axiom
Lebesgue_measure
Varying physical quantity that conveys information
classified into analog signals and digital signals; according to the determinacy of signals, classified into deterministic signals and random signals;
Signal
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
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
industry. These requirements are, in no particular order: Robustness, Determinacy, Compatibility. Robustness includes requirements such as connection redundancy
Process_control_network
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 Injector
PD
Fourth letter in the Greek alphabet
symbol of dentistry. The anonymous signature of James David Forbes. Determinacy (having a definite truth-value) in philosophical logic. In mathematics
Delta_(letter)
Finite ordered list of elements
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Tuple
Topics referred to by the same term
dead reckoning, GNSS-assisted dead reckoning for vehicles Axiom of real determinacy Azerbaijan Democratic Republic (1918–1920), a precursor state to modern
ADR
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
Topics referred to by the same term
computer-networking environment) Zermelo–Fraenkel set theory with the axiom of determinacy This disambiguation page lists articles associated with the title ZFD
ZFD
Field of engineering dealing with extremely low tolerances
precision Flexures Kinematic coupling Measurement uncertainty Kinematic determinacy This article incorporates public domain material from the National Institute
Precision_engineering
Term in mathematical logic
added hypothesis that ZF is consistent. The axiom of determinacy The axiom of real determinacy AD+ A set of sentences is independent, or simply independent
Independence (mathematical logic)
Independence_(mathematical_logic)
Clade of seed plants that produce flowers
ISBN 978-1-4398-4436-6. Bortiri, E.; Hake, S. (13 January 2007). "Flowering and determinacy in maize". Journal of Experimental Botany. 58 (5). Oxford University
Flowering_plant
Device which holds a mirror
This type of mount is designed according to the principles of kinematic determinacy. Typically, the movable frame that holds the mirror pivots on a ball
Mirror_mount
Axiom of set theory
that are not compatible with the axiom of choice, such as the axiom of determinacy. While some varieties of constructive mathematics avoid the axiom of
Axiom_of_choice
the axiom of choice. However, (1)-(4) is complete when extended with a determinacy schema for certain parity games. S2S can also be axiomatized by Π13 sentences
S2S_(mathematics)
Hand game for two players or more
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Rock_paper_scissors
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
Plant which completes its life cycle within one growing season and then dies
Rohde, A; Beeckman, T (2008). "Flowering-time genes modulate meristem determinacy and growth form in Arabidopsis thaliana". Nature Genetics. 40 (12): 1489–92
Annual_plant
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 set theory
the existence of 0 ♯ {\displaystyle 0^{\sharp }} is equivalent to the determinacy of lightface analytic games. In fact, the strategy for a universal lightface
Zero_sharp
Concept in mathematical logic
connection with strong partition cardinals and the axiom of determinacy. The axiom of determinacy is equivalent to the existence of unboundedly many strong
Theta_(set_theory)
strong partition cardinals contradicts the axiom of choice. The axiom of determinacy implies that ℵ1 is a strong partition cardinal. Henle, James M.; Kleinberg
Strong_partition_cardinal
Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy Axiom of real determinacy Empty set Forcing (mathematics)
List of mathematical logic topics
List_of_mathematical_logic_topics
Subfield of mathematics
Contemporary research in set theory includes the study of large cardinals and determinacy. Large cardinals are cardinal numbers with particular properties so strong
Mathematical_logic
Mathematical models of strategic interactions
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Game_theory
Concept in game theory
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Focal_point_(game_theory)
Proposition in mathematical logic
the axiom of choice (AC) (and therefore the negation of the axiom of determinacy, AD), so choice and GCH are not independent in ZF; there are no models
Continuum_hypothesis
are determined. Projective determinacy Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical
Homogeneously_Suslin_set
Existence of 0# Singular cardinals hypothesis Projective determinacy (and even the full axiom of determinacy if the axiom of choice is not assumed) There are
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Mathematical game played on a directed graph
for n = 2), where determinacy of such games was proven. The Knaster–Tarski theorem leads to a relatively simple proof of determinacy of parity games. Moreover
Parity_game
Search algorithm
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Alpha–beta_pruning
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
Hungarian and American mathematician and physicist (1903–1957)
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
John_von_Neumann
1994 mathematics book
Woodin cardinals. The book concludes with the chapter "Determinacy", involving the axiom of determinacy and the theory of infinite games. Reviewer Frank R
The_Higher_Infinite
French mathematician (1871–1956)
Borel–Cantelli lemma Borel–Carathéodory theorem Heine–Borel theorem Borel determinacy theorem Borel right process Borel set Borel summation Borel distribution
Émile_Borel
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
Proof all ranked voting rules have spoilers
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Arrow's_impossibility_theorem
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
Concept in game theory involving long-term strategic planning
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Farsightedness_(game_theory)
Model of conflict for two players in game theory
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Chicken_(game)
Mathematical logician and philosopher
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Kurt_Gödel
Type of perfect Bayesian equilibrium
Vickrey Combinatorial game theory Core concepts Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game
Separating_equilibrium
Polish-American mathematician (1932–2025)
sequence Mycielskian Mycielski–Grötzsch graph Mycielski's theorem Axiom of determinacy graph theory Awards Stefan Banach Prize (1965) Fellow of the American
Jan_Mycielski
the projective sets and below the sets in L(R). Assuming sufficient determinacy, the class of inductive sets has the scale property and thus the prewellordering
Inductive_set
Topics referred to by the same term
representation of a Lie group, abbreviated "Ad" in mathematics Axiom of determinacy, a set theory axiom Antiproton Decelerator, a device at the CERN physics
AD_(disambiguation)
Israeli mathematician and computer scientist (1931–2026)
n = 2) is decidable. A key component of the proof implicitly showed determinacy of parity games, which lie in the third level of the Borel hierarchy
Michael_O._Rabin
American mathematician (1934–2007)
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Paul_Cohen
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
German philosopher (1806–1856)
closely related to "Philosophy" than to "Religion", based on their shared determinacy and clarity, and a common ethical root. However, Stirner went beyond
Max_Stirner
Axiom of Zermelo-Fraenkel set theory
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Axiom_of_infinity
Rule established in an earlier legal case
members, Justices Scalia and Thomas seem to have the most faith in the determinacy of the legal texts that come before the Court. It should come as no surprise
Precedent
Set whose elements all belong to another set
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Subset
Fixture which constrains all degrees of freedom of a part
to be restrained and therefore are predictable. Kinematics Kinematic determinacy Precision engineering Slocum, Alexander (April 2010). "Kinematic Couplings:
Kinematic_coupling
One-to-one correspondence
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Bijection
Pair of logical equivalences
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
De_Morgan's_laws
Collection of mathematical objects
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Set_(mathematics)
Set theory concept
part II". Notices of the American Mathematical Society. 48 (7): 681–690. "Large Cardinals and Determinacy" at the Stanford Encyclopedia of Philosophy
Large_cardinal
Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set
Nested_set_collection
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
Person involved in directing, instructing and training sportspeople
Names Athletic coach, sports coach Activity sectors Physical education, determinacy Description Fields of employment Schools Related jobs Teacher, sports
Coach_(sport)
Algorithm similar to Gaussian elimination
equations in the problem Serial numbers of problems Number of problems Determinacy 2 2 2, 4, 5, 6, 7, 9, 10, 11 8 Determinate 3 3 1, 3, 8, 12, 15, 16 6
Fangcheng_(mathematics)
Natural philosophy holding that the world comprises fundamental indivisible components
be considered a composite of form and matter, as it has identity and determinacy to a certain extent, pure or primary matter is completely unformed, unintelligible
Atomism
Smallest transitive inner model of ZF containing all the ordinals and all the reals
L(R) does not satisfy the axiom of choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, given
L(R)
Difference of an open set by a meager set
\Gamma } has the property of Baire. Therefore, it follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective
Property_of_Baire
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