Gödel's proof

(redirected from Godel's incompleteness theorems)

Gödel's proof

[′gərd·əlz ′prüf]
(mathematics)
Any formal arithmetical system is incomplete in the sense that, given any consistent set of arithmetical axioms, there are true statements in the resulting arithmetical system that cannot be derived from these axioms.
References in periodicals archive ?
It was like fooling a kid into eating his vegetables," Professor Rosenhouse said in a telephone interview, adding, "Raymond took something like Godel's incompleteness theorems and used a string of logic puzzles as a device for presenting them.
Once students have mastered Hardy's version of the proof (Hardy, 1941; Padula, 2006) together with Arianrhod's (2003) highly explanatory version, and perhaps, the proof of the existence of an infinity of prime numbers (Padula, 2003), Hardy's (1941) other example of an 'elegant' proof, they can progress to more difficult proofs such as: Godel's incompleteness theorems (Padula, 2011) and the story of the solving of Fermat's last theorem (Singh, 2005).
1992, Godel's Incompleteness Theorems, Oxford University Press, Nueva York.
Further, from a technical perspective, by appealing to infinitary formal systems to which Godel's incompleteness theorems do not apply, neo-formalists clearly overcome Godelian worries.
We can often read that the unrealizability of Hilbert's program is proved by Godel's incompleteness theorems.
Let us now turn to Godel's incompleteness Theorems.
programs which recognize theorems) in that the former do not seem to be limited by Godel's incompleteness theorems whereas the latter do seem to be thus limited.
While this argument does support the view that mechanical theorem recognizers are limited by Godel's incompleteness theorem, the argument suffers from the following weakness: the argument does not show that each mechanical theorem recognizer is bound to some fixed set of axioms, but only that the theorems recognized could have been obtained alternatively by effective productions from such a set of axioms.