[FOM] Disproving Godel's explanation of incompleteness

Michael Kremer kremer at uchicago.edu
Tue Oct 18 14:52:07 EDT 2005

Roger Bishop Jones think that he can refute the explanation of the 
significance of the incompleteness theorems in terms of truth by defining a 
"truth" predicate for ZFC under the "provability semantics" where

p is true <--> p is provable in ZFC

p is not true <--> ~p is provable in ZFC.

But why should we call this a "truth" predicate, or suppose that Godel 
would have agreed to call it a "truth" predicate?  Note that not only is it 
a gappy predicate, it fails to satisfy certain obvious principles that a 
truth predicate should satisfy, among them:

p v q is true <--> p is true or q is true.

Godel, clearly, had in mind by a truth predicate a totally defined 
predicate satisfying all the instances of

p is true <--> p,

(Tarski's T-schema)

and of course if this is what a truth predicate is, then Godel's 
explanation has not been refuted.

But if we want to propose some alternative truth predicate not satisfying 
this account, we need some story about what makes a predicate a truth 
predicate.  And Jones hasn't provided that.

--Michael Kremer

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