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Math Help - Incompleteness thereom

  1. #1
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    Incompleteness 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?
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  2. #2
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    There are a couple of non-universal notations here. What is a "Regular Φ axiom system", namely what does Φ mean? And \Omega(\phi) is the formula stating consistency of \phi, something that is also denoted \text{Con}(\phi) or \text{Consis}(\phi)?

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