FOM: consis-completeness again
michael Detlefsen
Detlefsen.1 at nd.edu
Thu Feb 5 13:52:46 EST 1998
Let me take another pass at clarifying my completeness question. Sorry if
this is beginning to bore some of you. This time, I'll attempt
clarification by direct comparison of my question with Steve's. My view is
this: Steve's question is not mine, but it is something like a special case
of mine. I'll try explaining the differences by considering a particular
case.
Let GC stand for the sentence of PA that expresses Goldbach's Conjecture. I
choose GC because it is mathematically interesting and may be undecidable
in PA. W.r.t. GC, I want to know whether its being a theorem of PA would
make PA inconsistent. Steve, as I understand him, would put this question
as follows:
(*) Is it the case that PA |- GC--> Con(T)?
That's cool as far as I am concerned since this question is importantly
related to my question. Here's how.
Having a positive answer to (*) would show, in my parlance, that if GC were
provable in PA, then PA would be inconsistent. So, a positive answer to (*)
would answer that instance of my question having to do with GC. But I want
to know about more than just GC. So does Steve, as I understand him. He and
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.
My question goes farther. It would persist in the face of a negative answer
to (*). Why? Because PA's proving 'GC--> Con(T)' isn't the only conceivable
means by which GC's provability in PA might engender a contradiction in PA.
Such, at any rate, was my though when I originally framed the question.
Thus, in addition to Steve's question, I am also asking: Are there
sentences S of PA such that (i) PA does not prove 'S-->Con(T)', (ii) PA |-
GC--> S, and (iii) if PA |- S, then PA is inconsistent. And, again, I want
to know this not only of GC, but of all sentences that are undecidable in
PA.
Beyond this, I also want to know the answers to the parallel questions for
systems other than PA. And, to take the philosophical question motivating
my interest, I want ultimately to know whether the answers to these
questions make PA special in some way (i.e. whether it confers upon it an
interesting type of completeness which I called consistency-completeness).
Were PA to be complete in this sense, would that constitute a
foundationally interesting difference between it and various other theories
of foundational interest (e.g. ZF)? This latter, of course, we can think
about without having answers to the questions coming before it.
I hope that helps to clarify my question concerning
consistency-completeness. Now I'd like to say a little more about
counterfactual or subjunctive conditionals and why I view them as
indispensable to understanding Godel's theorems. I hope this addresses
points raised by Joe Shipman and Torkel Franzen. Joe and Torkel seem to
regard the conditional 'if G were provable in PA, then PA would be
inconsistent' as intolerably unclear. Joe says that to clarify it, I need
to reconceive it as a statement asserting the provability in some
particular well-defined system of the parallel material conditional. I
don't accept that. I am willing to say, of course, that the associated
material conditional can be formalized and proved in any number of known
systems (e.g. PA, ZF, etc.). When I assert it, however, I am asserting it
relative to what I know or believe and I don't have a formal specification
of that "system" (if, indeed, it is a system). I know, however, that if
(the logical closure of) my body of beliefs is a system that it is not
equivalent to either PA nor ZF. That's because I believe all the axioms of
both those systems and also believe in their consistency. So, when I assert
'if G were provable in PA, then PA would be inconsistent', I am not
asserting the provability of the material version of this conditional in
either PA or ZF or any other system of mathematical propositions of which I
am aware.
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.
Mic Detlefsen
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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1 at nd.edu
FAX: 219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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