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The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for a conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930. [28]
A humorous variant of Gödel's ontological proof is mentioned in Quentin Canterel's novel The Jolly Coroner. [26] [page needed] The proof is also mentioned in the TV series Hand of God. [specify] Jeffrey Kegler's 2007 novel The God Proof depicts the (fictional) rediscovery of Gödel's lost notebook about the ontological proof. [27]
Equiconsistency. In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not possible to prove the absolute consistency of a theory T.
ω-consistent theory. In mathematical logic, an ω-consistent (or omega-consistent, also called numerically segregative) [1] theory is a theory (collection of sentences) that is not only (syntactically) consistent [2] (that is, does not prove a contradiction), but also avoids proving certain infinite combinations of sentences that are ...
Gödel's completeness theorem. The formula (∀ x. R (x, x)) → (∀ x ∃ y. R (x, y)) holds in all structures (only the simplest 8 are shown left). By Gödel's completeness result, it must hence have a natural deduction proof (shown right). Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a ...
OCLC. 62532514. The General Theory of Employment, Interest and Money is a book by English economist John Maynard Keynes published in February 1936. It caused a profound shift in economic thought, [1] giving macroeconomics a central place in economic theory and contributing much of its terminology [2] – the "Keynesian Revolution".
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. [1] This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Existence of inaccessible cardinals. Existence of Mahlo cardinals. Existence of measurable cardinals (first conjectured by Ulam) Existence of supercompact cardinals. The following statements can be proven to be independent of ZFC assuming the consistency of a suitable large cardinal: Proper forcing axiom. Open coloring axiom.