Set Theory and Logic

Skolem Revisited - Thoralf's Achilles Heel

Authors: Hannes Hutzelmeyer
Comments: 12 Pages.

The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and a precise analysis of foundation of mathematics in general, including theory of types. A famous paper by Thoralf Skolem of 1934 is usually put at the beginning of publications on non-standard arithmetic. A critical investigation shows that it has serious, if not even insurmountable problems. Firstly one notices that it is based on second-order logic, it has unary and binary function-variables, and binary operator-constants (that map two functions to a function). It seems strange that one makes a fundamental statement about first-order logic systems using second order. In proving Satz 1 on the asymptotic behavior of arithmetic functions Skolem has some inaccuracies and formal errors. These minor problems can be solved by diligent work. But even if one has replaced his metalingual use of his relation symbols Bi by an ontologically correct method there remains secondly the problem of transitivity of the minority relation of functions that is neglected by Skolem. Thirdly, in constructing the strictly ascending function g of Satz 1 use is made of recursion by a dot-dot-dot notation. This is not an admissible procedure in object-language, although there may be a correct way to solve the problem in metalanguage. Therefore one does not only need second-order logic in combination with a precise object-language (in order to avoid ontology problems) but also a precise use of metalanguage (in order to avoid dot-dot-dot) for justifying Skolem's Satz 1 after one has eventually solved the transitivity problem; Skolem's Satz 2 would then be valid. However, as long as Satz 1 is not confirmed it does not pay to treat Satz 3 and Satz 4, leaving open the existence of non-standard models of arithmetic on the basis of Skolem's work
Category: Set Theory and Logic