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In [[mathematical logic]], '''Gödel's ''β'' function''' is a function used to permit quantification over finite ...tial step of that proof. Gödel gave the ''' ''β'' function''' its name in (Gödel 1934). ...

{{Distinguish|text=the [[Gödel Prize]]}}{{Short description|Award in mathematical logic}} ...red the second annual Gödel Lecture, entitled ''Will Logicians be Replaced by Machines?''}}</ref> ...

...aleph_1}</math>, of the [[Continuum Hypothesis]] (CH). It was discussed by [[Nikolai Luzin]] in 1935, although he did not claim to be the first to pos ...y the [[Continuum Hypothesis]], which follows from [[V=L]]. It is implied by [[Martin's Axiom]] together with the negation of the CH.{{r|EOM}} ...

...oms|Peano arithmetic]] because the latter can't see the truth value of its Gödel sentence, while [[Mind|human minds]] can. Mathematician [[Roger Penrose]] m ...not necessarily defined (there is no implicit notion of "truth"). However, Gödel's statement is actually true in the standard model of natural numbers. See ...

...y came up with the first ontological argument" (p. 22).</ref> was proposed by [[Anselm of Canterbury|Saint Anselm of Canterbury]] in his 1078 work, ''[[P ...Other arguments have been categorised as ontological, including those made by Islamic philosophers [[Mulla Sadra]] and [[Allama Tabatabai]]. ...

{{quote|This is a 1 followed by about three hundred thousand zeroes ... Even if such a machine were to oper ...one, some, or all of the possible sequences of nondeterministic decisions, by enumerating states after each possible decision. ...

{{defn|no=2|defn=[[Ω-logic]] is a form of logic introduced by Hugh Woodin}} {{defn|defn=''s''⁀''x'' is the extension of a sequence ''s'' by ''x''}} ...

...of [[geometry]] in terms of analysis, and the invention of [[set theory]] by [[Georg Cantor|Cantor]], eventually led to the much more general modern con ...pringer Science & Business Media|isbn=978-3-540-65340-0|pages=154–}}</ref> By 1718, he came to regard as a function "any expression made up of a variable ...

...quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehen ...pt]], earliest known [[decimal|decimal system]] allows indefinite counting by way of introducing new symbols.<ref name="buffalo1">{{cite web|url=http://w ...

...ther. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common ...nce, so the philosophy of logic investigates philosophical problems raised by logic.<ref name="Haack1">{{cite book |last1=Haack |first1=Susan |title=Phil ...

...r example, complex propositions are made up of simpler propositions linked by [[Logical connective|logical vocabulary]] like <math>\land</math> ([[Logica ...idered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century m ...