Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
Hemiolia and Escher In Verbum (Fig. 149), oppositions are made into unities on several levels. Going around we see gradual trans ...
FIGURE 53. Three Spheres II, by M. C. Escher (lithograph, 1946). Indra's Net Finally, consider Three Spheres II (Fig. 53), in wh ...
one for each person. Each one would contain calls to many others, thus creating a virtual cloud of A TN's around each A TN. Call ...
We made two crucial observations in Chapter I: (1) that the MU-puzzle has depth largely because it involves the interplay of len ...
I-count. The reason is that if 3 divides 2n, then-because 3 does not divide 2-it must divide n (a simple fact from the theory of ...
It is easy. Clearly this mapping b(~tween notations is an information- preserving transformation; it is like playing the same me ...
tions changing one pattern of symbols into another, or as a series of arithmetical operations changing one magnitude into anothe ...
(1) 31 gIven (2) 311 rule 2 (m= 1, n= 1) (3) 31111 rule 2 (m=2, n=ll) (4) 301 rule 3 (m= 1, n= 1, k=3) (5) 3010 rule 1 (m=30) (6 ...
and then "arithmetize" its rules, we could call the producible numbers "pq-numbers"-and so on. Note that the producible numbers ...
The Dual Nature of MUMON In order to gain some benefit from this peculiar transformation of the original question, we would have ...
Codes and Implicit Meaning Now it could be objected here that a coded message, unlike an uncoded message, does not express anyth ...
will have to Godel-number TNT itself, just as we did the MIU-system, and then "arithmetize" its rules of inference. The Godel-nu ...
626,262,636,626,262,163,636,362,262,112,123,262,163,323,111,123,362,262,112,262,163,323 axiom 3 Va:Va (a+Sa')=S(a+a') 626,262,16 ...
arithmetized MIU-system, only there are more rules and axioms, and to write out arithmetical equivalents explicitly would be a b ...
We can dignify this observation by calling it the Central Dogma of Mathematical Logic, and depicting it in a two-step diagram: T ...
methods of reasoning, and therefore that TNT never has falsities for theorems. In other words, anything which is a theorem of TN ...
PART II ...
Prelude • • • Achilles and the Tortoise have come to the residence of their friend the Crab, to make the acquaintance of one of ...
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He immediately realized that this equation has infinitely many solu- tions a, b, c, and then wrote in the margin the following n ...
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