Axiom of choice - Independence of AC - Article Index

Encyclopedia - Axiom Of Choice: Encyclopedia - Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originall...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring Choice In Intuitionistic Logic, Though Not Classically
Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is no...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Independence Of Ac
By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (Z...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Usage
Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set X contains...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring Choice In Intuitionistic Logic Though Not Classically
Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is no...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Statement
The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X. Stated more formally: Let X...   » Read the article

Encyclopedia - Axiom Of Choice: Encyclopedia Ii - Axiom Of Choice - Results Requiring ¬ac
There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory pl...   » Read the article