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A statement of number theory is not about a statement of number theory it just is a statement of number theory…Godel had the insight that a statement of number theory could be about a statement of number theory (possibly even itself) if only numbers could somehow stand for statements…a code…numbers are made to stand for symbols…this coding trick enables statements of number theory to be understood on two different levels as statements of number theory, and as statements about statements of number theory…”This statement of number theory does not have any proof”…”G is unprovable…but true”.’ Whole numbers are not statements, nor are their properties. Historically it was thought that, ‘ statements are about properties of whole numbers. Therefore if you had a system from which you could derive anything, that system would not be useful to us. I think the reason we are supposed to limit ourselves to consideration of ‘consistent’ systems, is that ‘anything follows from a contradiction’, that to construct an inconsistent system would be trivial and pointless, to the extent that the purpose of one of these formal systems is to derive things from them.
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Hofstadter’s first summation of Godel reads, ‘All consistent axiomatic formulations of number theory include undecidable propositions’.
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I’ve been through this book a couple times before, but I’m still not entirely sure that I have the Godel stuff down pat, so this time I will take it slow, and I’ll write it all down here to make sure my brain/mind isn’t fooling itself into unwarranted confidence about its familiarity with the material. Hofstadter: ‘Without doubt, Strange Loops involving rules that change themselves, directly or indirectly, are at the core of intelligence.’ His paradigm case of a strange loop is Godel’s Incompleteness Theorem, the introduction of the Bach and Escher seems to be mostly a combination of elucidation, repetition, isomorphic restatement of main points for rhetorical purposes, and perhaps a little bit of literary catering.