Newsgroups: comp.ai.philosophy
Path: cantaloupe.srv.cs.cmu.edu!rochester!udel!news.mathworks.com!uhog.mit.edu!grapevine.lcs.mit.edu!olivea!uunet!psinntp!scylla!daryl
From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: rereRe: The end of god
Message-ID: <1994Nov3.203014.2198@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Thu, 3 Nov 1994 20:30:14 GMT
Lines: 75

rickert@cs.niu.edu (Neil Rickert) writes:

>jqb@netcom.com (Jim Balter) writes:
>
>>Suppose there is a consistent axiomatic system S with sufficient
>>expressibility to encode the proposition "S is consistent".  Then, by
>>Goedel, that proposition cannot be proved within S.  Yet it most
>>certainly is true and is not false.
>
>Whether one can argue this depends on one's theory of truth.

No, it doesn't. Jim started by saying "Suppose there is a consistent
axiomatic system S". Whatever view of truth you are using, surely
the truth of "S is consistent" follows from the assumption "S is
consistent".

>However, if one takes the position that truth only has a meaning
>within a formal system, then your argument is not persuasive.

I would say that truth of a statement is only meaningful given an
*interpretation* of the terms in the statement. However, in the case
Jim is talking about, I believe that it is clear that the
interpretation of "is consistent" in the statement "S is consistent"
is to be the same interpretation of "is consistent" as in the
assumption "Suppose that S is a consistent axiomatic system."

>One cannot say "it most certainly is true".  Instead, one can only say
>"In the formal system S' it most certainly is true."

Formal systems don't determine the truth of statements, they determine
the provability of statements.

>>               It can be proved in some system greater than S.
>
>If there is also another system greater than S in which it can be
>disproved, this argument loses much of its persuasiveness.

If S is consistent, then there is no formal system encoding and
proving the proposition that S is inconsistent. This is a tricky
point. Jim started with the assumption:

    Suppose there is a consistent axiomatic system S with sufficient
    expressibility to encode the proposition "S is consistent".

You need to ask what it *means* to say that one formal system S' can
encode the proposition that another formal system S is consistent (S'
may be the same system as S or a different one). Usually, that is
taken to mean the following:

     1. There is a way to code statements of S as terms of S'.
     2. There is a one-place predicate P(x) definable in S'
        such that for every statement A in the language of S,
        S' proves P(#A) if and only if S proves A (where #A is the
        code of A).

These conditions imply that provability in S can be represented in S'.
Given these conditions, S' can encode "S is consistent" as:

        con(S) == not P(#false)
where false is any logical contradiction.

Now, by condition 2. if provability in S is representable in S', then
S' proves P(#false) if and only if S proves false. That shows that S'
cannot prove S is inconsistent unless S really is inconsistent.

That doesn't mean that I can't add the axiom P(#false) ("S is
inconsistent") to S and get a new consistent theory S'. However, in
this new theory, P no longer represents provability in S, so it is
not really correct to say that S' proves that S is inconsistent.

Daryl McCullough
ORA Corp.
Ithaca, NY


