Unknow052106 - 1 REMARKS ON THE UNKNOWABLE by Harvey M....

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1 REMARKS ON THE UNKNOWABLE by Harvey M. Friedman Ohio State University Gödel Centenary April 28, 2006 revised May 21, 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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2 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. H(t+1,x) = f(H(t,x-1),H(t,x),H(t,x+1)). In H(t,x), t 0 is time, and x Z is position. H(t,x) is the state at time t and position x.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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Unknow052106 - 1 REMARKS ON THE UNKNOWABLE by Harvey M....

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