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Mathematical logic

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Set theory and paradoxes: "remaining
In 1910, the first volume of ''[[Principia Mathematica]]'' by Russell and [[Alfred North Whitehead]] was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of [[type theory]], which Russell and Whitehead developed in an effort to avoid the paradoxes. ''Principia Mathematica'' is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics ([[#CITEREFFerreirós2001|Ferreirós 2001]], p. 445).
Fraenkel ([[#CITEREFFraenkel1922|1922]]) proved that the axiom of choice cannot be proved from the remaining axioms of Zermelo's set theory with [[urelements]]. Later work by [[Paul Cohen (mathematician)|Paul Cohen]] ([[#CITEREFCohen1966|1966]]) showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of [[forcing (mathematics)|forcing]], which is now an important tool for establishing [[independence result]]s in set theory.<ref>See also {{harvnb|Cohen|2008}}.</ref>
==== Symbolic logic ====