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REMARKS ON THE UNKNOWABLE
by
Harvey M. Friedman
Ohio State University
Gödel Centenary
April 28, 2006
revised May 20, 2006
I will talk about a specific candidate for unknowability. I
have attempted to make this candidate as widely interesting
as I can on short notice.
A number of formal results and conjectures have emerged,
and it appears that the approach here has opened up some
new lines of research.
I will focus attention on mathematical candidates for
unknowability. There is, of course, the wider topic of the
unknowability of propositions involving physical objects or
other kinds of nonmathematical objects.
The kind of unknowability I will discuss concerns
the count of certain
natural finite sets of objects.
Even the situation with regard to our present strong formal
systems is rather unclear. One can just profitably focus on
that, putting aside issues of general unknowability.
Many of the ideas presented here are present in work of
Chaitin, although in a different form. We haven’t looked at
the overlap. In particular, we propose that our transition
systems is a particularly good vehicle for developing these
ideas. Also the idea of exploiting special features of the
standard axiom systems used for the foundations of
mathematics, in this context, seems novel.
TRANSITION SYSTEMS
A 1 dimensional transition system, 1TS, is given by a
quadruple (S,a,b,f), where
1. S is a finite set.
2. a,b S, a ≠ b.
3. f:S
3
S.
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The evolution of (S,a,b,f) is given by H:N Z
S, where
H(0,x) = a if x < 0; b o.w.
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 Fall '08
 JOSHUA
 Math, Formal language, Finite set, Metalogic, Gödel's incompleteness theorems, ZFC

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