[FOM] Fwd: invitation to comment
Thilo Volker Weinert
weinert at math.uni-bonn.de
Sun May 22 10:09:20 EDT 2011
Just a little comment here:
I personally do not doubt the consistency of PA, neither the one of ZFC. Yet I disagree on the statement that the structure of ε_0 would be well-understood. Attempts to clarify the status of partition relations like ε_0 --> (ε_0, n)^2 for all natural n have hitherto been unsuccessful and to my mind, understanding the structure would necessitate such a clarification.
Thilo Weinert
Am 21.05.11 08:47, schrieb Vaughan Pratt:
>
>> 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 theorem.
>
> Vaughan Pratt
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