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Chapter 8

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Overview Edit

Typographical Number Theory

An extension of the Propositional Calculus called "TNT" is presented. In TNT, number-theoretical reasoning can be done by rigid symbol manipulation. Differences between formal reasoning and human thought are considered.

While you read Edit

Questions Edit

  1. Why does DRH prefer the two translations as given on p. 210 of the sentence "1729 is a sum of two cubes."?.
  2. On p.212 there are 6 well-formed formulas of TNT. For each of them state what they mean and whether they are true or false. (Hint: Either four of them are true and two false, or four false and two true.)
    1. ~∀c:∃b:(SS0·b)=c
    2. ∀c:~∃b:(SS0·b)=c
    3. ∀c:∃b:~(SS0·b)=c
    4. ~∃b:∀c:(SS0·b)=c
    5. ∃b:~∀c:(SS0·b)=c
    6. ∃b:∀c:~(SS0·b)=c
  3. On p.215 there are some translation exercises, translate the first four into TNT-sentences and the last into an open well-formed formula. The sixth is tricky and DRH warns to try this "... only if you are willing to spend hours and hours on it - and if you know quite a bit of number theory!"
    1. All natural numbers are equal to 4.
    2. There is no natural number which equals its own square.
    3. Different natural numbers have different successors.
    4. If 1 equals 0, then every number is odd.
    5. b is a power of 2.
    6. b is a power of 10.
  4. Translate the five Peano postulates on p.216 by replacing the words Genie, djinn and meta into the more common form.
  5. Use the axioms and the rules up to and including p. 218 to produce the theorem ~∀b:∃a:Sa=b
  6. What's wrong with the derivation on p.220? What is a valid derivation?
  7. Translate Peano's fourth postulate into TNT-notation, and then derive that string as a theo

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