The fundamental aim of the paper is to
correct a harmful way to interpret a Godel’s erroneous remark at the Congress
of Konigsberg in 1930. Although the Godel’s
fault is rather venial, its misreading has
produced and continues to produce dangerous fruits, so as to apply the incompleteness Theorems to the full second-order Arithmetic and to
deduce the semantic incompleteness of its language by these same Theorems. The
first three paragraphs are introductory and serve to define the languages inherently semantic and its properties,
to discuss the consequences of the expression order used in a language and some
questions about the
semantic completeness. In particular, it is
highlighted that a non-formal theory may be semantically complete despite using
a language semantically incomplete. Finally, an alternative interpretation for the Godel’s unfortunate comment is proposed.
Turing,
A.M. (1937) On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical
SocietyII, 42, 230-265. http://dx.doi.org/10.1112/plms/s2-42.1.230
Godel, K. (1931)
On Formally Undecidable Propositions of Principia Mathematica and Related
System I, doc. 1931. In: Feferman, et al., Eds., Kurt Godel Collected Works,
Vol. I, Oxford University Press, Oxford, 195.
Henkin, L. (1949)
The Completeness of the First-Order Functional Calculus. The Journal of Symbolic Logic, 14, 159- 166. http://dx.doi.org/10.2307/2267044
Skolem, T. (1933)
über die Unmoglichkeit..., Norsk
matematisk forenings skrifter, Series
II, 10, 73-82. Reprinted in: Fenstad, J.E.
(1970) Selected Works in Logic, Oslo University, Oslo, 345-354.
Feferman, et al. (1995) op. cit. Volume III, Oxford University Press, Oxford, 439. In the
textual notes is written: “The copy-text for *1930c [...] was one of several items in an
envelope that Godel labelled “Manuskripte Korrekt der 3 Arbeiten in
Mo[nats]H[efte] Wiener Vortrage über die ersten zwei” (manuscripts, proofs
for the three papers in Monatshefte [1930, 1931, and 1933i] plus Vienna
lectures on the first two.) On the basis of that label, *1930c ought to be the
text of Godel’s presentation to Menger’s colloquium on 14 May 1930—the only
occasion, aside from the meeting in Konigsberg, on which Godel is known to have
lectured on his dissertation results [...]. Internal evidence, however,
especially the reference on the last page to the incompleteness discovery,
suggests that the text must be that of the later talk. Since no other lecture
text on this topic has been found, it may well be that Godel used the same
basic text on both occasions, with a few later additions”.
Godel, K. (1931)
doc.1931. In: Feferman, et al., Eds., op. cit., Volume I, Oxford University Press, Oxford, 187. The
purpose is simply to be able to express formulas such as Vx(f(x)), where f is a recursive function, which
strictly are not permitted in the first-order classical Logic.
Von Neumann, J. (1925)
An Axiomatization of Set Theory. In: Heijenoort, V., Ed., From Frege to Godel, Harvard University
Press, Cambridge, CA, 412. Here is an especially interesting remark by Von Neumann: “[...] no
categorical axiomatization of set theory seems to exist at all; [...] And since
there is no axiom system for mathematics, geometry, and so forth that does not
presuppose set theory, there probably cannot be any categorically axiomatized
infinite systems at all”.
Maltsev, A. (1936) Untersuchungen
aus dem Gebiete der mathematischen Logik. Matematicheskii Sbornik, 1, 323-336. In: Maltsev, A. (1971) The
Mathematics of Algebraic Systems: Collected Papers 1936-1967, Amsterdam.
