From: bevans@carina.unm.edu (Mathemagician)
Newsgroups: alt.atheism
Subject: Re: Undecidability and the FAQ
Date: 9 Sep 92 12:31:44 GMT
Organization: Society for the Preservation of E. coli

In article <TORKEL.92Sep7145557@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <=lqn!#f@lynx.unm.edu> bevans@carina.unm.edu (Mathemagician) writes:
>   >You just did.  Did you not just say that it is not provable in ZFC
>   >that the Continuum Hypothesis is undecidable in ZFC.  Is that or is
>   >that not claiming "that CH isn't known to be undecidable"?

>  Of course it isn't, unless one takes the view, which I do not, that
>nothing not provable in ZFC is known to be true.

Never said nor implied any such thing.  I said we are assuming ZFC to
be consistent because to assume otherwise is of no use.

This doesn't mean that we "know" ZFC is consistent.  However, things
that are "known" are claimed because of the assumption that our axioms
are consistent.  Things can be "known" to be false..."known" meaning
the truth or falsity of a statement can be resolved.

> In fact, in claiming
>to know that it is not provable in ZFC that CH is undecidable in ZFC,
>I am implicitly claiming to know that ZFC is consistent, which is not
>provable in ZFC either.

Of course.  That's the Second Incompleteness Theorem.

So?

>  Although it has all been said before, I repeat the simple points
>involved.

I disagreed with these "simple points."  You never gave an explanation
as to why you felt my disagreements are in error.

>  The statement "A is undecidable in T" implies that the theory T is
>consistent, whatever the statement A may be. This follows from the
>definition of "A is undecidable in T" as meaning "neither A, nor the
>negation of A is provable in T". A theory (using classical logic) in
>which not everything is provable is consistent.

I disagree with your first sentence.  The statement "A is undecidable
in T" doesn't "imply" that T is consistent...it assumes it is.  While
it may certainly be true that a theory is consistent if and only if it
contains undecidable statements, that doesn't mean we are looking at
the if part of that implication.  Since we assume that the theory is
consistent before we make any statements, the only if part of the
implication becomes irrelevant.  Our antecedent, "T is consistent" is
not capable of having the value of false.  Therefore, p <=> q (which
is the case regardless of p, for this case) becomes p => q (where p is
not allowed to be false since we are assuming it is true).

>  In particular, in ZFC it is provable that "if CH is undecidable in
>ZFC, then ZFC is consistent".

Yes, but we aren't dealing with that problem.  ZFC is already assumed
to be consistent...which would allow that statement to be deemed true
in the first place.

>  Hence, by Godel's second theorem, (assuming the consistency of ZFC),
>it is not provable in ZFC that CH is undecidable in ZFC.

Godel's Second Theorm of Incompleteness is that it is undecidable
whether or not the axioms are consistent.  Thus, we assume they are
which allows us to claim that certain statements are undecidable.

I'm dealing with pragmatics.  An inconsistent axiom system is of no
use, so we disallow that possibility, a priori.  We know that, if we
allow the inconsistency of the axioms, p <=> q.  But if we disallow
the inconsistency of the axioms, p ~<= q.

If we assume that the axioms are consistent, we make claims that
certain statements are undecidable.

Yes, it is undecidable that the axioms are consistent which implies
that our undecidable statements are undecidable.

However, an inconsistent axiom system is of little use so we disallow
that possibility.  In practical use, it means that we make tenable
claims that certain statements are undecidable.  It is, essentially, a
"zeroth axiom" that the axioms are consistent.  Axioms aren't there
because they can be proven true.  They are there because it is
"reasonable" to assume them to be true.  It is "reasonable" to assume
that the axioms are consistent since to have an inconsistent axiom
system is to be unable to clearly prove things.

Thus, assume the axioms are consistent.

-- 
Brian Evans                |     "Bad mood, bad mood...Sure I'm in a bad mood!
bevans@carina.unm.edu      |      I haven't had sex...*EVER!*" -- Virgin Mary
Newsgroups: alt.atheism