By Howard DeLong

**Publish 12 months note:** First released in 1971

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This textual content explores the historic purposes for the formation of Aristotelian good judgment, the increase of mathematical good judgment, the character of the formal axiomatic approach and its use, and the most result of metatheory and their import.

*From 1971 edition*

Includes 22 figures and 19 tables. Appendixes. Bibliography. Indexes.

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More precisely, by substitution of some of the preceding functions at the argument places of one of the preceding functions, for example, Not all variables on the left side need occur on the right side (the same applies to the recursion schema (2)). 28. We include classes among relations (as one-place relations). Recursive relations R, of course, have the property that for every given n-tuple of numbers it can be decided whether R(x1,…, xn) holds or not. 29. Whenever formulas are used to express a meaning (in particular, in all formulas expressing metamathematical propositions or notions), Hilbert’s symbolism is employed.

E. that: (2) The formula Consis Ap(p) is provable. Gödel planned to establish (2) in detail in a second part of the paper; but it seems no one doubted it. Hilbert and Bernays in Volume 2 of Grundlagen der Mathematik (1939) carried out this exercise in a similar system. Let us accept (2). Then Consis cannot be provable in the system, if the system is simply consistent (Gödel’s second incompleteness theorem). For, if Consis were provable, then from it and (2) by modus ponens, Ap(p) would be provable; and we have seen that Ap(p) is not provable if the system is simply consistent.

For, if Consis were provable, then from it and (2) by modus ponens, Ap(p) would be provable; and we have seen that Ap(p) is not provable if the system is simply consistent. This is the second way in which Gödel’s paper (1931) showed that Hilbert’s formalist programme cannot be carried out in any simple way. Hilbert hoped, after formalizing the core of classical mathematics, to show the resulting system consistent by what he called in German ‘ﬁnit’ means, which in my book (1952) I translated as ‘ﬁnitary’.