When1: 1930
When2: 1939
Who: Kurt Gödel [Gödel, Kurt]
What: mathematician/logician
Where: Czech Republic/Slovakia/USA
works\ On Formally Undecidable Propositions of Principia Mathematica and related systems [1931]; Consistency of the Axiom of Choice and of the Generalized Continuum-hypothesis with the Axioms of Set Theory [1940]
Detail: He lived 1906 to 1978. First-order predicate calculus and first-order logic are complete [1930]. All formal arithmetic systems must be incomplete [1931]. For all formal and consistent arithmetic systems, at least one true arithmetic proposition cannot be formally decidable. Neither proposition nor negation has proof, so arithmetic system is incomplete {Gödel's first incompleteness theorem}. Propositions are statements about numbers. Propositions have Gödel-number codes. Systems have propositions about propositions, and at least one such statement is not provable, because proofs use self-referential number statements. Therefore, it is impossible to prove system consistency using arithmetic.
Formal or logical systems are logically equivalent to recursively definable functions and arithmetic systems. Computing machines embody such functions. Therefore, machines can never prove their consistency or completeness.
The continuum hypothesis is consistent with basic set-theory axioms [1938 to 1939].
Epistemology
Definitions can specify class elements and their relations, and relations can make new elements {recursive definition}.
Mathematical objects and concepts are real and separate from mind. People know fundamental mathematical truths by intuition.
Mathematical Sciences>Mathematics>History>Axiomatic Theory
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Date Modified: 2022.0224