Foundations
The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logic; symbolic logic). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets, originated by Georg Cantor, which now constitutes a universal mathematical language.
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