Axiom of real determinacy
- Axiom of real determinacy
In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following:
:Consider infinite two-person games with perfect information. Then, every game of length ω where both players choose real numbers is determined, i.e., one of the two players has a winning strategy.
The axiom of real determinacy is a stronger version of the axiom of determinacy, which makes the same statement about games where both players choose integers; it is inconsistent with the axiom of choice. ADR also implies the existence of inner models with certain large cardinals.
ADR is equivalent to AD plus the axiom of uniformization.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Axiom of determinacy — The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two person games of length ω with perfect information. AD states that every such game in… … Wikipedia
Determinacy — Determined redirects here. For the 2005 heavy metal song, see Determined (song). For other uses, see Indeterminacy (disambiguation). In set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other… … Wikipedia
Axiom of choice — This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of … Wikipedia
Proper forcing axiom — In the mathematical field of set theory, the proper forcing axiom ( PFA ) is a significant strengthening of Martin s axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement A forcing or partially… … Wikipedia
List of axioms — This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self evidence. Individual axioms are almost always part of a larger axiomatic… … Wikipedia
List of mathematical logic topics — Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic … Wikipedia
List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A … Wikipedia
List of set theory topics — Logic portal Set theory portal … Wikipedia
L(R) — In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals. It can be constructed in a manner analogous to the construction of L (that is, Gödel s constructible universe), by … Wikipedia
Independence (mathematical logic) — In mathematical logic, a sentence sigma; is called independent of a given first order theory T if T neither proves nor refutes sigma;; that is, it is impossible to prove sigma; from T , and it is also impossible to prove from T that sigma; is… … Wikipedia
