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Published online by Cambridge University Press: 22 January 2016
For an investigation on the foundations of mathematics to get an adequate mutual understanding, it is necessary to describe the generalities of the investigation with the parallel description of its particularities. Although one should obtain an exact and precise knowledge only through the latter, it is almost impossible, without the former, to get the underlying ideas and the fundamental principles, upon which the investigation is based.
This is the Part (VII) of my papers with the same major title. The knowledge of Part (I) and (II) (Hamburger Abhandlungen and of Part (IV) (Nagoya Math. J. vol. 13 (1958)) are assumed.
2) Cf. the beginning of the introduction in Part (II).
3) Cf. § 10 for the universal constant V.
4) Cf. the introduction in Part (I) and also §5, Part (I).
5) Cf. Chapter V, Part (II) and also Introduction in Part (I).
6) Cf. Part (X), forthcoming.
7) Cf. § 11, Part (I).
8) Appearing in this same volume.
9) All formulas except *N*1 are irreducible, namely the proofs given below are all irreducible except that for *N*1. But the irreducibility is not our concern at present.
10) Particular attention is required when a formula 
obtained from a formula Fa
, previously proved, by replacing a by a dependent variable p is used as cut formula of an ordinary cut in a proof P. For, if the variable a is used in the proof of Fa
as a substitute for some bound variable, then p is a set in the proof of 
accordingly also in the proof P. If this is not the case, then p is not necessarily a set in P, since the cut formula 
is proved without using p as set. For instance, the cut formula of the cut N*10 in the proof of N*ll is the formula obtained from N*10 by replacing a by a′ and a is used in the proof of N*10 as set at the right above of the proof, since a is substituted for the element variable u of the set P for induction. Therefore a′ is a set in the proof of N*11.
The variable P in N*5 is an independent variable.
11) Let T be an axiomatic theory in UL, which is assumed to be consistent and to contain the natural number theory to some sufficient extent. Then by Gödel’s theorem there is a formula A in T which is unprovable in T and intuitively holds. Then, as was stated by Gödei, we get a consistent axiomatic theory T′ by adjoining the negation of A to the system of axioms of T, and in T′ some theorems, like 
, contradict our intuitive knowledge. However, T′ is not an axiomatic theory in UL, unless 
has been proved in some consistent extension of T in UL. It is, therefore, an open problem, whefher there is a consistent axiomatic theory in UL, which includes the natural number theory to sufficient extent, and in which a provable formula contradicts our intuitive knowledge.
12) Cf. for instance, N. Bourbaki: Élements de mathématique, Première partie, Livre III. Chapitre I, Structures topologiques, Paris, 1951, pp. 9, 10.
13) The quantifier 
for the element variable u is omitted.
14) In the proof of (8) the set p is used only for “understanding”.
15) Cf. Gesammelte mathematische Werke, Braunschweig, 1932, vol. III, p. 357. Cf. also Bolzano: Paradoxien des Unendlichen, Leibzig, 1851 § 13, which is mentioned by Dedekind, ibid, in the foot note.
16) Cf. §3, Part (VIII), appearing in this same volume.
17) Cf. Part (III), Nagoya Math. J. vol. 13 (1958).
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