FOM: consis-completeness again
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
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!
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