FOM: consis-completeness again
Arnon Avron
aa at math.tau.ac.il
Sat Feb 7 03:36:08 EST 1998
> I want to know (at a minimum) of every sentence S of PA that is not
> decidable in PA whether PA |- S--> Con(T). So, I want an answer to Steve's
> question ... but that's not all I want.
The Rosser Sentence R(T) is a counterexample (this has explicitely
been shown in my paper: "A note on provability, truth and existence",
JPL 20: 403-409, 1991, but should have been obvious in any case)
Moreover: if A is any sigma_0_1 sentence which is not decidable
in T then A--> Con(T) is NOT provale in T (again the trivial proof
is given in my paper cited above. The claim is equivalent to
theorem 5.2.1 in Smorynski chapter in the Handbook of Mathematical Logic).
> I believe, moreover, that some such vantage is necessary for a full grasp
> of Godel's theorems. Perhaps a story will explain why. Let Ralph be someone
> who believes the material conditional 'if PA is consistent, then G is not
> provable in PA' and believes that PA is consistent. Suppose, for
> simplicity, that Ralph's beliefs are logically closed. Ralph thus believes
> that G is not provable in PA.
>
> Now, suppose Ralph is asked the following question:
>
> (?*): Ralph, would you still believe in PA's consistency were you to find
> out that G were provable in PA?
>
> If Ralph were to say 'yes', I would say that he does not have what I would
> count as knowledge of Godel's theorem. He must answer 'no' if I am to give
> him credit for understanding the proof of the theorem. (This is exactly
> what I would say of a student in Ralph's position. Would anyone give a
> relevantly different answer?) Hence, I want to say further that having a
> proof of Godel's theorem must do more than merely provide a warrant for
> asserting the material conditional 'if PA is consistent, then G is not
> provable in PA'. It must as well give one at least enough sense of
> 'necessity' to say that 'if G were to be provable in PA, then PA would be
> inconsistent'. Hence, I believe that grasp of some counterfactual or
> subjunctive conditional is necessary for genuine knowledge of Godel's
> theorems.
I fail to understand the argument here. If someone knows
the material conditional 'if PA is consistent, then G is not
provable in PA' and that G IS provable in PA then classical
logic immediately forces her to recognize that PA is inconsistent
(using the disjunctive syllogism). No understanding of Goedel's PROOF
is needed for this!
Arnon Avron
Computer Science Department
Tel-Aviv University
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