Monday, November 14, 2005

Godel

Godel's theorem as it appeared in Proposition VI of his 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I.":

To every w-consistent recursive class k of formulae there correspond recursive class-sings r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(k) (where v is the free variable of r).

which translates to

All consistent axiomatic formulations of number theory include undecidable propositions.
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Godel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.
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2 comments:

amar said...


All consistent axiomatic formulations of number theory include undecidable propositions.

I also like the following:

Any axiomatic system of number theory is either incomplete or inconsistent.

sj said...

Copied this over from Amar's blog.
An year of incompleteness