Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
You may recognize Peano's fifth postulate as the principle of mathe- matical induction-another term for a hereditary argument. P ...
back the universal quantifier on theorems which contain variables that became free as a result of usage of specification. Acting ...
Va:-Sa=O 3b:Va:-Sa=b aXiom existence You might now try to shunt symbols, according to rules so far given, to produce the theorem ...
Illegal Shortcuts Now here is an interesting question: "How can we make a derivation for the string O=O?" It seems that the obvi ...
Something Is Missing Now if you experiment around for a while with the rules and axioms of TNT so far presented, you will find t ...
there was no way to express general statements about addition in its sym- bolism, let alone prove them. The equipment simply was ...
It doesn't jibe with what we believe about addition. But it is one possible extension of TNT, as we have so far formulated TNT. ...
pattern is a theorem in itself. That is like proving that each djinn passes a message to its meta, as in the children's game of ...
This is the first of the two input theorems required by the induction rule. The other requirement is the first line of the pyram ...
(26) (d+SO)=(Sd+O) (27) Vd:(d+SO)=(Sd+O) (28) Vc:Vd:(d+Sc)=(Sd+c) transitivity (lines 23,25) generalization induction (lines 18 ...
(54) (c+O)=(O+c) (55) Vc:(c+O)=(O+c) [0 commutes with every c.] * * * * * (56) Vd:Vc:(c+d)=(d+c) transitivity (lines 50,53) gene ...
Formal Reasoning vs. Informal Reasoning I would have preferred to show how to derive Euclid's Theorem (the infinitude of primes) ...
pIe, if TNT were complete, number theorists would be put out of businesf>: any question in their field could be resolved, wit ...
needed in proving that such a typographical property holds. In other words, if properties of integers are not used-or if only a ...
A M u Offering 1 The Tortoise and Achilles have just been to hear a lecture on the origins of the Genetic Code, and are now drin ...
whole Oriental mysticism trip, with the I Ching, gurus, and whatnot. So one day I'm thinking to myself, "Why not Zen too?" And t ...
Tortoise: Hmm ... Now THERE'S something I could like about Zen. Achilles: I had a hunch you'd get attached to it. Tortoise: But ...
Achilles: Well, there is one problem. Although both are widely told koans, my master has cautioned me that only one of them is g ...
FIGURE 45. La Mezquita, by M. C. Escher (bln,ck and white chalk, 1936). apparent, but from special angles, beautiful regularity ...
look like (picks up a nearby napkin, and draws for the Tortoise these four figures): oococo Tortoise: They are mysterious-lookin ...
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