[FOM] Fwd: invitation to comment

Vaughan Pratt pratt at cs.stanford.edu
Sat May 21 02:47:12 EDT 2011

> From: Vladimir Voevodsky <vladimir at ias.edu>
> Date: May 18, 2011 4:44:13 PM EDT
> To put it very shortly I think that in-consistency of Peano arithmetic
> as well as in-consistency of ZFC are open and very interesting problems
> in mathematics. Consistency on the other hand is not an interesting
> problem since it has been shown by Goedel to be impossible to proof.

This last sentence of Voevodsky makes very clear that he does not 
understand Goedel's second incompleteness theorem, which states only 
that the consistency of T cannot be proved *in T*.  That the consistency 
of PA cannot be proved in PA might mean something if PA were a powerful 
theory, but it is not.  As Martin pointed out, consistency of PA can be 
proved in PRA + ε_0.  The structure of ε_0 is well understood, and 
anyone claiming otherwise needs to show why.  Dismissing ε_0 with a wave 
of the hand as Voevodsky did is not mathematics, it is just handwaving 
based seemingly on a misunderstanding of Goedel's second incompleteness 

Vaughan Pratt

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