From: bevans@carina.unm.edu (Mathemagician)
Subject: Re: Undecidability and the FAQ
Date: Fri, 28 Aug 92 11:59:03 GMT
Organization: Society for the Preservation of E. coli

In article <1992Aug27.111517.7800@sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <wjcn78h@lynx.unm.edu> bevans@carina.unm.edu (Mathemagician) writes:
>   >Well, I've found a bit of time to re-subscribe to alt.atheism, and I
>   >went through my brain-child, the FAQ.

>  Hello there, I see you're back and going on about the continuum
>hypothesis as usual. As I have pointed out before, your comments,
>apart from having no apparent relevance to atheism, are incorrect.

Oh, lord...here we go again.

If it has nothing to do with atheism, why is undecidability in the FAQ?

>   >"Aleph-one" is
>   >arrived at by taking 2 to the power of aleph-null.  "Aleph-two" by
>   >taking 2 to the power of aleph-one, etc.

>  Aleph one is not at all defined as 2 to the power of aleph null, but
>as the smallest infinite cardinal greater than aleph null. The assertion
>that aleph one is 2^aleph null is the continuum hypothesis.

>  But why go on about this? You are apparently wedded to your
>"mathemagical" mangling of set theory.

Why go on about this?  Because the Continuum Hypothesis, however I
might misstate, is known to be undecidable.  Yes, as I go through my
Real Analysis book, I see that the power function of generating a
series of transfinite numbers isn't defined as the aleph series...and
that the Continuum Hypothesis is that there is no transfinite number
between aleph-null and c (as defined in _Analysis:  An Introduction to
Proof_).

However, that still doesn't change the fact that the Continuum
Hypothesis is known to be undecidable given the Zermelo-Fraenkel
axioms of set theory.

>  As for the FAQ file entry, your comments are irrelevant, since what
>is asserted in the entry is that it is undecidable "in the formal
>system" whether an undecidable statement A is undecidable. I didn't
>attack this formulation, since it is indeed true that if Godel's
>second theorem applies to a theory T, then it cannot be proved in T
>for any A that A is undecidable in T.

Incorrect.  It is quite decidable "in the formal system" whether an
undecidable statement A is undecidable.

From _Analysis:  An Introduction to Proof_:

Whether it is true or false is still an unanswered--perhaps
unanswerable--question.  It is known, however, that the asusmption of
the continuum hypothesis does not contradict any of the usual axioms
of set theory.  (This was proved by Kurt Godel in 1938.)  But lest one
take too much comfort in this, we should also point out that it has
been proven (by Paul Cohen in 1963) that the denial of the continuum
hypothesis does not lead to any contradictions either.

Thus the continuum hypothesis is undecidable on the basis of the
currently accepted axioms for set theory.  (It can neither be proved
nor disproved.)  It remains to be seen whether new axioms will be
found that will enable future mathematicians finally to settle the
issue.

From _Mathematics:  The New Golden Age_:

Some progress was made when, in 1938, Kurt Godel used new techniques
of mathematical logic to show that from the Zermelo-Fraenkel axioms it
is definitely not possible to prove that the set of real numbers does
not have size aleph-one.  But this did not solve the problem, since it
was still possible that the axioms were simply not sufficient to
decide the issue one way or the other.

Despite this possibility, however, in the years following Godel's
result there was a widely held expectation that the continuum problem
was in fact decidable within the Zermelo-Fraenkel framework.  In which
case, since Godel had shown that the answer was definitely not
provable to be anything other than aleph-one, it had (it was assumed)
to be the case that the continuum did have size aleph-one, and that
given time this would eventually be proved conclusively.  Accordingly,
it did not seem at all unreasonable to assume this anticipated result
whenever a piece of mathematics required a knowledge of the size of
the continuum, and a grat many resutls were proved on the assumption
that the continuum hypothesis (as the assumption was known) was true.

And then, in 1963, came the announcement that Paul Cohen of Stanford
Universtiy had developed a new logical technique by which he had been
able to prove that the continuum hypothesis could not be deduced from
the Zermelo-Frankel axioms.  When combined with the earlier result of
Godel, this showed that the continuum hypothesis was in fact
undecidable in the Zermelo-Fraenkel system.

So, Torkel, I defy you again.  Given a formal system, it *is* possible
to deduce if a statement A is undecidable within the framework of that
system.  The Continuum Hypothesis in the Zermelo-Fraenkel system is one
such statement.

What does this have to do with atheism?  Well, in the section about
Godel's Theorem (point 21), it leaves the impression that questions of
the existence of supernatural entities might be undecidably
undecidable.  That is, that the entire point for this group is
meaningless as we won't even be able to know that we won't be able to
know (brain in a vat images come springing to mind.)

I find that to be a very important thing.  Personally, I find the idea
of a axiomatic theory complex enough to include integer arithmetic
capable of having anything to say about supernatural entities a crock.
But there are people out there who do think it's possible...and that's
why the point is in the FAQ.

-- 
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
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