現代数学の系譜11 ガロア理論を読む25 [無断転載禁止]©2ch.net

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274: 現代数学の系譜11 ガロア理論を読む [sage] 2016/11/12(土) 08:17:11.54 ID:CRbt3jrT

>>274 つづき

Relationship to the incompleteness theorem

Godel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. The name for the incompleteness theorem refers to another meaning of complete (see model theory ? Using the compactness and completeness theorems).

It shows that in any consistent effective theory T containing Peano arithmetic (PA), the formula CT expressing the consistency of T cannot be proven within T.

Applying the completeness theorem to this result, gives the existence of a model of T where the formula CT is false. Such a model (precisely, the set of "natural numbers" it contains) is necessarily non-standard, as it contains the code number of a proof of a contradiction of T. But T is consistent when viewed from the outside.
Thus this code number of a proof of contradiction of T must be a non-standard number.

In fact, the model of any theory containing PA obtained by the systematic construction of the arithmetical model existence theorem, is always non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive (as recursive definitions would be unambiguous).

Also, there is no recursive non-standard model of PA.

(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1477804000/274

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