Lemma: Suppose that φ is a Regular Φ axiom system that is consistent. Then φ must be unable to prove the sentence Ω(φ).

Can someone explain to me why Ω(φ) must be a true sentence?

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- Apr 27th 2010, 03:23 PMjjbrianIncompleteness thereom
Lemma: Suppose that φ is a Regular Φ axiom system that is consistent. Then φ must be unable to prove the sentence Ω(φ).

Can someone explain to me why Ω(φ) must be a true sentence? - Apr 28th 2010, 11:15 AMemakarov
There are a couple of non-universal notations here. What is a "Regular Φ axiom system", namely what does Φ mean? And is the formula stating consistency of , something that is also denoted or ?

Presumably, says something "there is no -derivation of 0=1". If it is false (in the standard model ), then there exists, in fact, a -derivation of 0=1. Then is contradictory and therefore derives everything. However, it is proved that does not prove .