A Source Book in Mathematics: volume 1 by David Eugene Smith

By David Eugene Smith

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Everything else is only a façon de parler. Hilbert’s Program consists in establishing, by metamathematical means, that general mathematics10 are conservative over recessive formulas. A result proven by « infinite » methods, can, modulo transformations – not necessarily very friendly, but possible « in principle » – be established in a strictly finitary setting. This is therefore a principle of purity of methods, taken from the practice of number theory (elementary proofs), but which Hilbert tried to mechanise.

To avoid systemism, it is important to observe that polarity comes from the idea of a « logical onion », of which we peal the successive skins. Thus the opposition answer/question seem to recover the opposition expansive/recessive. On the other hand, positive/negative only relates to the most external skin and, in the same way, explicit/implicit seems to only make sense on the external layer. 1 The system RR. The first incompleteness theorem is usually formulated for systems based upon a simplistic language, due to Peano.

13 W. , but he draws no consequence, because he is a bit slow. On the other hand, V. is very smart and made his PhD on the Baghdad cuckolds, so he thinks: « Gosh, if I were not a cuckold, W. would have concluded that it’s him and killed his wife ». Therefore V. slays his innocent spouse. The moral: too much epistemic logic can damage your health. What is the point of turning this gag topsy-turvy? It illustrates the difference between constatation and deduction. W. must perform a simple deduction, but he is a moron and is unable to reach a conclusion.

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