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...

game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass. This, combined with the Borel determinacy theorem of Martin...

closed set, then the game is determined. By the Borel determinacy theorem, games whose winning set is a Borel set are determined. It follows from the existence...

theorem Heine–Borel theorem Borel determinacy theorem Borel right process Borel set Borel summation Borel distribution Borel's conjecture about strong measure...

Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical...

Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and...

projective determinacy, all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC proves Borel determinacy...

theorem for random fractals". Mathematics. 10 (5): 706. doi:10.3390/math10050706. hdl:1807/110291. Leinster, Tom (23 July 2021). "Borel Determinacy Does...

replacement are required to show that Borel games are determined. The proven result is Donald A. Martin's Borel determinacy theorem. A later, more careful analysis...

{\displaystyle S} is a Borel subset of Baire space is determined. Every infinite cardinal κ satisfies 2×κ = κ. Measure theory The Vitali theorem on the existence...

independently proves the theorem a second time. 1898 Bernstein's proof (not relying on the axiom of choice) is published by Émile Borel in his book on functions...

Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets...

that Brouwer's theorem is equivalent to the determinacy theorem for Hex. The Lefschetz fixed-point theorem says that if a continuous map f from a finite...

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...

this chapter, and the proof in it by Donald A. Martin of the Borel determinacy theorem, as central for Kanamori, "a triumph for the theory he presents"...

Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy Axiom of real determinacy Empty set Forcing (mathematics)...

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...

ring and order theory (see Boolean prime ideal theorem).[citation needed] However, the axioms of determinacy and dependent choice together are sufficient...

\Gamma } has the property of Baire. Therefore, it follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective...

Universally measurable set Determinacy AD+ Axiom of determinacy Axiom of projective determinacy Axiom of real determinacy Empty set Forcing (mathematics)...

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...

principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses...

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...

Sometimes also called "rudimentary set theory". BC Berkeley cardinal BD Borel determinacy Berkeley cardinal A Berkeley cardinal is a cardinal κ in a model of...

acceptance by a non-deterministic automaton. Determinacy: Provably in ZFC, Borel games are determined, and the determinacy proof for boolean combinations of Π02...

example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Let P ( α ) {\displaystyle P(\alpha )} be a property defined for...

familiar with differential and integral calculus, and by twelve he had read Borel's La Théorie des Fonctions. He was also interested in history, reading Wilhelm...

The number of elements in each equivalence class is the natural number n. Borel equivalence relation Cluster graph – Graph made from disjoint union of complete...

book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix...

a construction is called a Borel code. Given a Borel set B {\displaystyle B} in V {\displaystyle V} , one recovers a Borel code, and then applies the...