Kurt Gödel developed a deductive system, applicable for both finite- and infinite-valued first-order logic (a formal logic in which a predicate can refer to a single subject) as well as for intermediate logic (a formal intuitionistic logic usable to provide proofs such as a consistency proof for arithmetic), and showed in 1932 that logical intuition cannot be characterized by finite-valued logic. He generalized the system to many-valued logics in 1922 and went on to develop logics with ℵ 0 (infinite within a range) truth values. Jan Łukasiewicz developed a system of three-valued logic in 1920. However, the definition of a random real number, meaning a real number that has no finite description whatsoever, remains somewhat in the realm of paradox. Felix Hausdorff demonstrated the logical possibility of an absolutely continuous ordering of words comprising bivalent values, each word having absolutely infinite length, in 1938. Richard Dedekind, who defined real numbers in terms of certain sets of rational numbers in the 19th century, also developed an axiom of continuity stating that a single correct value exists at the limit of any trial and error approximation. Isaac Newton and Gottfried Wilhelm Leibniz used both infinities and infinitesimals to develop the differential and integral calculus in the late 17th century.
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