Curious. Isn't the proof of Godel's theorem a special case of this? As I understand it, the proof is this:
Consider the statement: This theorem is not provable. If it is false, the theorem is provable. Since 'provable' implies true, this is a contradiction. Therefore the theorem is true, which means it is true and not provable. The genius in Godel's method is that he created an isomorphism between the domain of the previous paragraph, and arithmetic, and the isomorphism preserves truth and provability. Thus the above theorem corresponds to a statement in arithmetic that is true and not provable. What is this statement, you might ask. Well, evidently it is far to complex to compute or write down (although it would be interesting to see if more powerful computers or quantum computers would change this.) Anyway, that true but non-provable theorem shows that number theory (aka arithmetic) is incomplete -- that's the definition of incomplete in this context. --Barry On Apr 16, 2013, at 10:25 AM, Owen Densmore <o...@backspaces.net> wrote: > On Sat, Apr 13, 2013 at 2:05 PM, Nicholas Thompson > <nickthomp...@earthlink.net> wrote: > Can anybody translate this for a non programmer person? > > Nick's question brings up a project I'd love to see: an attempt at an > isomorphism between computation and philosophy. (An isomorphism is a 1 to 1, > onto mapping from one to another, or a bijection.) > > For example, in computer science, "decidability" is a very concrete idea. > Yet when I hear philosophical terms, and dutifully look them up in the > stanford dictionary of philosophy, I find myself suspicious of circularity. > > Decidability is interesting because it proves not all computations can > successfully expressed as "programs". It does this by using two infinities > of different cardinality (countable vs continuum). > > Does philosophy deal in constructs that nicely map onto computing, possibly > programming languages? > > I'm not specifically concerned with decidability, only use that as an example > because it shows the struggle in computer science for modeling computation > itself, from Finite Automata, Context Free Languages, and to Turing Machines > (or equivalently lambda calculus). > > I don't dislike philosophy, mainly thanks to conversations with Nick. And I > do know that axiomatic approaches to philosophy have been popular. > > So is there a possible isomorphism? > > -- Owen > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College to unsubscribe http://redfish.com/mailman/listinfo/friam_redfish.com
