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From: jfh@browngr.UUCP (John "Spike" Hughes)
Newsgroups: net.math
Subject: Re: Beyond Exponentiation
Message-ID: <1837@browngr.UUCP>
Date: Fri, 8-Feb-85 10:26:43 EST
Article-I.D.: browngr.1837
Posted: Fri Feb  8 10:26:43 1985
Date-Received: Mon, 18-Feb-85 04:44:07 EST
References: ihnet.186
Lines: 20


    The super-exponential is an interesting object--Ed Nelson (of Princeton)
has been giving talks about fundamentals of mathematics--logic and such--
questioning whether, without the induction axioms, certains things
are 'computable' oe even 'expressible'. Super-exponentiation of
large numbers seems not to be 'expressible' in his terms. If
you can catch him giving a lecture on this, you should go--when he
gave it at Haverford last year itwas called "Do the integers exist?"
It's thought provoking.

   By the way, as for defining S-exponentiation, you might want to
look at how the Gamma Function (an extension of factorial) was
built. In Calculus, by Michael Spivak, there is a mention of
a cxondition which uniquely defines the gamma function, indicating
what a 'nice' interpolation it is: gamma(x) = (x-1)! for x an \
integer (positive), and gamma is smooth on its domain, and
log(gamma(x)) is convex, which in some sense describes the extreme
smoothness of gamma. You might want to try a similar interpolation
mechanism...
  -jfh