FOM: Re: theory-edge mailing list, tautologies

[V.Z. Nuri]

hi all. thanks for some fresh perspectives.
on theory-edge we have almost no one
who is able to intelligently interpret
FD's work, partly because of its 
abstruse sophistication, partly because of the
logic background. we're enthusiastic &
gradually getting some of it. but
neither do we take him totally
seriously either, haha

some references have been flying
to "merely arithmetical statements". these
statements are supposedly not affected
by the continuum hypothesis. on the other
hand, we know there are important results
in math that depend on whether CH
is assumed true or false. (yes, I did
skim that sci am article when it came
out in the 90s, <wink>)

so I am wondering if there is a way
of characterizing these kinds of statements
(in particular/detail, what is 
"non merely arithmetical?")
such that a nonspecialist might stand
a chance of recognizing them when
he sees them.

it seems to me that maybe any "complex
enough" set of axioms may lead to CH-like
situations in which important results
can be taken either way (both the
assertion & the negation). and I am wondering
if they may exist in complexity theory,
which is now a very large edifice of
theory.

of course all this doesnt make sense if
one agrees that all of computational
complexity theory is just built out
of ZF. on the other hand it seems to
me computational complexity theory has
not yet been "axiomatized" in the sense
of being reduced to a set of axioms.

and when this was attempted for
math at the beginning of this century,
it was found, contrary to intuition,
various thms did not
yield so easily. maybe complexity theory
might be similar. until it is 
rigorously axiomized by someone, 
I am not so certain. (another part
of me scoffs that there should be any
important aspect of complexity theory
like P=?NP that is not provable.)

by the way FD recently posted a very
short bio to our list.. he's posted
preliminary drafts to our public archives
also.

http://groups.yahoo.com/group/theory-edge/message/3486

I like dr.schindler's stmt that if P=?NP
is independent of ZF,
"revolutionary methods would be called 
for"..imho thats always the best science..!!
I didnt understand why he says,
"it would be great if P=?NP is independent of ZF"..
because it would indicate how difficult or
remarkable the statement is, and require
brand new mathematics?

by the way, please correct me if I am
wrong, I am getting the impression
that "large cardinal hypotheses" are 
a trendy thing to work in and cutting
edge stuff in logic-- so for anyone to
relate any other (famous)
conjectures to "large cardinal
hypotheses" would immediately draw
significant attention.

such a motivation could characterize some
of FD's other/earlier work if you substitute
the term "unprovable" for "large cardinal
hypotheses".. on the other hand, arguably
he successfully carried out that agenda
with his prior/established results.

p.s. the archives look extremely intriguing
here and I think I am going to poke through
them a bit over time. 
my sincere thanks to dr.simpson for making
them publicly/conveniently accessable.


--- Martin Davis <martin at eipye.com> wrote:
> At 04:01 PM 7/31/2001 +0100, Roger Bishop Jones wrote:
> >In response to <JoeShipman at aol.com> Tuesday, July 31, 2001 4:45 AM
> >
> >| The Continuum Hypothesis or its negation cannot have anything to say about
> >| ... any ... mathematical statements
> >| which can be formulated arithmetically.
> >
> >Can you give a brief explanation for the non-specialist of why this is the
> >case?
> 
> Suppose A is an arithmetic statement and that either CH --> A or -CH --> A 
> is provable from the Zermelo-Fraenkel axioms (ZF). In the model of ZF found 
> by G\"odel in which CH is true, each sentence in the language of set theory 
> is interpreted relative to that  model. BUT ARITHMETIC SENTENCES ARE 
> ABSOLUTE - MEANING THAT THEIR INTERPRETATION IN THE MODEL IS JUST THE 
> STATEMENT ITSELF. So from CH --> A and modus ponens one gets A, relative to 
> the model, and therefore, A itself. For -CH --> A, similar considerations 
> apply to the models Cohen found in which -CH is ture.
> 
> Martin


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