He lived 1843 to 1930 and studied geometry foundations [1882], especially line and point interchangeability.
He lived 1858 to 1932. He invented logical notation, which Russell used. He studied axiomatic number systems. He invented Peano's postulates about rational numbers, based on Dedekind's work. He used reflexive, symmetric, and transitive axioms to derive rational numbers from natural numbers.
He lived 1880 to 1960 and axiomatized geometry using ideas of point and order.
He lived 1871 to 1952 and invented line and space axiomatic systems, building from points to lines to space. The three complete-quadrilateral diagonal points are never collinear {Fano's axiom}.
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.
He lived 1900 to 1982. Mathematics branches become more formal over time, until they are deductive systems. Mathematics is about deductive systems.
For axiomatic theories, subsets can use only some original terms but contain same theorems {Craig's theorem}.
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Date Modified: 2022.0225