Re: Gödel's theorems in Wikipedia

David C. Ullrich wrote: 
On 22 Jun 2005 15:53:56 -0700, ps218@xxxxxxxxx wrote:


Very nicely done!  A question arising: what's the nicest reference for
a proof of "first order arithmetic (Peano arithmetic or PA for short)
can prove that the largest consistent subset of PA is consistent."?



That can't be the question you meant to ask - it doesn't require
PA to prove that "the largest consistent subset of PA is consistent".

(First, the question seems to assume that any theory has a largest consistent subset, which is not so. Second, if a theory
does have a largest consistent subset then that subset is
consistent just by definition.)

The largest consistent subset of PA is defined with reference to a Gödel numering. Specifically, order the axioms of PA according to their Gödel numbers and take the largest initial segment of these axioms which does not lead to contradiction. Since PA is in fact consistent, this segment will in fact contain all of the axioms, but PA can't prove this.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

Relevant Pages

  • Re: =?ISO-8859-1?Q?G=F6del=27s_theorems_in_Wikipedia?=
    ... order the axioms of PA according to their Gödel numbers and take the largest initial segment of these axioms which does not lead to contradiction. ... "The largest consistent subset" would contain every consistent subset, and there's no reason that this should have that property. ... It's the largest consistent initial segment. ...
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  • Re: =?ISO-8859-1?Q?G=F6del=27s_theorems_in_Wikipedia?=
    ... As I explained to David Ulrich, the largest consistent subset of PA is defined as the largest initial segment of the axioms of PA ordered according to their Gödel numbers which does not lead to contradiction. ... Any sensible theory can prove that if an initial segment of some sequence of sentences does not lead to contradiction, the initial segment does not lead to contradiction. ... - Ludwig Wittgenstein, Tractatus Logico-Philosophicus. ...
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  • Re: Gödels theorems in Wikipedia
    ... >David C. Ullrich wrote: ... >> largest consistent subset, which is not so. ... It's the largest consistent initial segment. ...
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