Axiom of real determinacy
Encyclopedia
In mathematics
, the axiom of real determinacy (abbreviated as ADR) is an axiom
in set theory
. It states the following:
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 integer
s; it is inconsistent with the axiom of choice. ADR also implies the existence of inner model
s with certain large cardinals.
ADR is equivalent to AD plus the axiom of uniformization.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the axiom of real determinacy (abbreviated as ADR) is an axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. It states the following:
- Consider infinite two-person games with perfect informationPerfect informationIn game theory, perfect information describes the situation when a player has available the same information to determine all of the possible games as would be available at the end of the game....
. Then, every game of length ωOrdinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
where both players choose real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s 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
Axiom of determinacy
The axiom of determinacy 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...
, which makes the same statement about games where both players choose integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s; it is inconsistent with the axiom of choice. ADR also implies the existence of inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
s with certain large cardinals.
ADR is equivalent to AD plus the axiom of uniformization.