= 27d567bc2d48d9f40e9ccc20ea370b15)
Eliezer_Yudkowsky comments on How to cheat Löb's Theorem: my second try - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (25)
(Marcello observed this while reading the previous cheat attempt.) Since "1=0" contains no mentions of K, and is thus invariant under substitution of K-1 for K, why doesn't this language contain "provability of '0=1' implies 0=1" and thus prove 0=1?
Because the axiom contains an additional condition: it's "(K>0 and provability of '0=1') implies 0=1". Since you can't prove K>0, you can't apply Löb's theorem.
Now, you could add an axiom "K=12" and get a new sound proof system which would prove PPT.2 consistent; or you could substitute 12 for every not-quoted occurrence of K in the axioms of PPT.2, and get a new sound proof system which would prove PPT.2 consistent; but that's not a problem, because in both cases you get a new proof system, and there's nothing unusual about being able to construct a new system in which the old system's consistency can be shown.