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In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words...

be extended to an ultrafilter, but the proof uses the axiom of choice. The existence of a nontrivial ultrafilter (the ultrafilter lemma) can be added...

full strength of choice. Instead, it relies on the intermediate Ultrafilter principle. Using this theorem with the subbase for R {\displaystyle \mathbb...

former is equivalent in ZF to Tarski's 1930 ultrafilter lemma: every filter is a subset of some ultrafilter. One of the most interesting aspects of the...

In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is...

(as corollaries of Lindemann–Weierstrass) Tychonoff's theorem (to do) Ultrafilter lemma Ultraparallel theorem Urysohn's lemma Van der Waerden's theorem...

ultrafilter is called the ultrafilter lemma and cannot be proven in Zermelo–Fraenkel set theory (ZF), if ZF is consistent. Within ZF, the ultrafilter...

construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed. Vladimir Kanovei and...

Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann...

huge cardinal, then a strong form of Vopěnka's principle holds in Vκ: There is a κ-complete ultrafilter U such that for every {Ri: i < κ} where each Ri...

which it is also equivalent) Every proper filter is contained in an ultrafilter, a result that yields the completeness theorem of first-order logic In...

{\displaystyle \mathbb {R} ^{\mathbb {N} }} , as follows. Take a nonprincipal ultrafilter F ⊆ P ( N ) {\displaystyle F\subseteq P(\mathbb {N} )} . In particular...

denoted here S(B), called its Stone space. The points in S(B) are the ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean...

theories Transfer principle – Concept in model theory Ultrafilter – Maximal proper filter Leinster, Tom (2013). "Codensity and the ultrafilter monad" (PDF)...

(LNC) (also known as the law of contradiction, principle of non-contradiction (PNC), or the principle of contradiction) states that contradictory propositions...

("maximally deep"). The ultrafilter lemma The following important theorem is due to Alfred Tarski (1930). The ultrafilter lemma/principle/theorem (Tarski) — Every...

assumption that all objects satisfy the negation of the property. The principle may be formally expressed as the propositional formula ¬¬P ⇒ P, equivalently...

extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets...

All these definitions of Boolean algebra can be shown to be equivalent. Principle: If {X, R} is a partially ordered set, then {X, R(inverse)} is also a...

Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann...