I should correct myself. The mapping is not necessarily an isomorphism. --Barry
On Apr 16, 2013, at 3:39 PM, Barry MacKichan <barry.mackic...@mackichan.com> wrote: > 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
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