Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
contradiction in mathematics was found, mathematicians would im- mediately seek to pinpoint the system responsible for it, to ju ...
FIGURE 42. "Crab Canon", by M. C. Escher (-1965). ...
Tortoise: Achilles: Tortoise: Achilles: * * * * * * * * * * * * * ---* * ---* ---* * ---* ---* ---* * ---* ---* ---* ---* * ---* ...
Achilles: I don't know. But one thing for certain is that I don't worry about arguments of taste. De gustibus non est disputandu ...
FIGURE 43. Here is a short section of one of the Crab's Genes, turning round and round. When the two DNA strands are un- raveled ...
CR...J\5 CANON Sc5fj dc,~ WOlJ)j\:J a~5L::J oLJL.l -.-- ~I. I r-,.... "- (^01) L-I-- ~ ~ ~ - -....., ..- '- 0 I 0 ,..0 ~ v u - - ...
Tortoise: Achilles: Oh, really? I guess there's nothing better for you than walking. And it's a perfect day for a walk. I think ...
CHAPTER VIII Typographical Number Theory The Crab Canon and Indirect Self-Reference THREE EXAMPLES OF indirect self-reference ar ...
Now it may seem that we will need a symbol for each notion such as "prime" or "cube" or "positive"-but those notions are really ...
zero: one: two: three: etc. o 50 550 5550 The symbol 5 has an interpretation--"the successor of". Hence, the in- terpretation of ...
(SO+(SSO+SSSO)) ((SO+SSO) +SSSO) The next notion we'll symbolize is equals. That is very simple: we use '='. The advantage of ta ...
One way of changing an open formula into a closed formula, or sentence, is by prefixing it with a quantifier-either the phrase " ...
subject", such as "so-and-so". A string with free variables is like a predicate with "so-and-so" as its subject. For instance, ( ...
3e:SSSSSSO = (SSO· e) Now these three translations of "6 is even" are quite different strings, and it is by no means obvious tha ...
which asserts that 7 is a sum of two positive cubes. This will be just like the preceding sentence involving 1729, except that w ...
(b+SSO), but there is a shorter way to write it-namely, SSb. Likewise, "e plus 2" can be written SSe. Now, our translation is ex ...
Vc:3b:-(SSO·b)=c -3b:Vc:(SSO·b)=c 3b:-Vc:(SSO·b)=c 3b:Vc:-(SSO·b)=c (Second hint: Either four of them are true and two false, or ...
VARIABLES. a is a variable. If we're not being austere, so are b, e, d and e. A variable followed by a prime is also a variable. ...
This completes the table of Rules of Formation for the well-formed for- mulas of TNT. A Few More Translation Exercises And now, ...
which can be derived in the same way as was derived. Before we give more rules, let us give the five axioms of TNT: AXIOM 1: Va ...
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