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#artificial-intelligence #geb #goedel-escher-bach #hofstadter
So far, the only way we have found to represent prime numbers typographically is as a negative space. Is there, however, some way—I don't care how complicated—of representing the primes as a positive space—that is, as a set of theorems of some formal system? Different people's intuitions give different answers here. I remember quite vividly how puzzled and intrigued I was upon realizing the difference between a positive characterization and a negative characterization. I was quite convinced that not only the primes, but any set of numbers which could be represented negatively, could also be represented positively. The intuition underlying my belief is represented by the question: "How coulda figure andits ground not carry exactly the same information?" They seemed to me to embody the same information, just coded in two complementary ways. What seems right to you? It turns out I was right about the primes, but wrong in general. This astonished me, and continues to astonish me even today. It is a fact that: There exist formal systems whose negative space (set of non- theorems) is not the positive space (set of theorems) of any formal system
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  • owner: piotr.wasik - (no access) - Douglas Hofstadter "Goedel, Escher, Bach", p72
  • owner: naraypv - (no access) - Gödel, Escher, Bach_ An Eternal Golden Braid-Basic Books (1994).pdf, p80


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