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From: spector@suvax1.UUCP (mitchell spector)
Newsgroups: sci.lang,comp.ai
Subject: Re: Infinite alphabets - (Turing via Berke)
Message-ID: <758@suvax1.UUCP>
Date: Sun, 18-Oct-87 17:11:18 EDT
Article-I.D.: suvax1.758
Posted: Sun Oct 18 17:11:18 1987
Date-Received: Mon, 19-Oct-87 02:06:04 EDT
References: <557@nitrex.UUCP>
Organization: Seattle University, Seattle, WA.
Lines: 45
Xref: mnetor sci.lang:1589 comp.ai:925

In article <557@nitrex.UUCP>, rbl@nitrex.UUCP ( Dr. Robin Lake ) says:
 
> ....  If you go back to the fundamentals
> of math (Robinson's axiomatic set theory) and remove the Axiom of Choice, one
> can derive a new mathematics which is more appropriate for situations where
> uncertainty prevails.  This line of reasoning has worked extremely well in
> developing some powerful tools for "smart" data analyzers where the data is
> too dirty and uncertain for classic statistical techniques.
 
   What specifically are you thinking of?  Robinson developed non-standard
analysis, which is basically calculus with infinitesimals, done rigorously.
The axiom of choice *is* generally used to construct a hyperreal number
system (a structure containing both real numbers and infinitesimals which
has certain desirable properties), as in Robinson's theory.

   While there is a substantial body of mathematics in which the axiom of
choice is rejected, two things should be pointed out.  The first is that it
is almost never satisfactory merely to remove the axiom of choice from set
theory (or even to adopt as an axiom the negation of the axiom of choice;
the resulting theory is far too weak to prove many interesting results.
Instead, various strong axioms which happen to contradict the axiom of choice
are used as replacements; the chief of these is the axiom of determinateness.

   The second is that the acceptance or rejection of the axiom of choice has
no known effect on any practical problem (for example, anything in physics).
(This is certainly true if one is willing to adopt the much weaker axiom of
dependent choices, as almost everyone in the field does.)  For example, the
same sentences of number theory can be proven with the axiom of choice as
without it.  Moreover, all statements of calculus or analysis of the type
used in physics can be rephrased as statements of number theory, although
this can get rather tedious and technical.  It follows that none of these
statements require the axiom of choice for their proofs either.  It is only
when one starts to consider things at a more abstract mathematical level that
the axiom of choice comes into play (for example, the existence of a
non-measurable set of real numbers).

   I'd be interested in the examples referred to in the previous message
(although I don't know if this discussion belongs in sci.lang -- that depends
on the examples, I guess).

-- 
Mitchell Spector
Dept. of Computer Science/Software Engineering, Seattle University
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