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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]
The following four independence results are also due to Gödel/Cohen.); the generalized continuum hypothesis (GCH); a related independent statement is that if a set x has fewer elements than y, then x also has fewer subsets than y. In particular, this statement fails when the cardinalities of the power sets of x and y coincide;
The consistency of the ... Gödel discovered a proof for the independence of the axiom of ... of Anselm of Canterbury's ontological proof of God's ...
The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of Grundlagen der Mathematik , along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full ...
In proof theory, the Dialectica interpretation [1] is a proof interpretation of intuitionistic logic (Heyting arithmetic) into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic.
Detlefsen (1990) argues that Gödel's theorem does not prevent a consistency proof because its hypotheses might not apply to all the systems in which a consistency proof could be carried out. [6] Dawson (2006) calls the belief that Gödel's theorem eliminates the possibility of a persuasive consistency proof "erroneous", citing the consistency ...
Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent computably enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own ...
Ward defended the utility of the five ways (for instance, on the fourth argument he states that all possible smells must pre-exist in the mind of God, but that God, being by his nature non-physical, does not himself stink) whilst pointing out that they only constitute a proof of God if one first begins with a proposition that the universe can ...