Tags:axiom of chpoce, foundations of mathematics, Löwenheim–Skolem theorem, proper and improper interpretations and Skolem's paradox
Abstract:
The great Norwegian scientist, mathematician and logician Thoralf Skolem (1922) introduced the “relativity of the notion of set” and the “improper interpretation” of a set-theory structure, based on the axiom of choice. One can demonstrate that they can serve as a ground of the logic of ground. The thesis is: The “improper interpretation” of an infinite set-theory structure founds the “proper interpretation” and thus that structure self-founds itself as the one interpretation of it can found the other. Furthermore, the interesting corollaries are implied about the theory of information for information can be introduced as a mapping of the proper into the improper interpretation of a mathematical structure being also a quantity of its complexity in the sense of Kolmogorov, Chaitin, and Martin-Löf (1965-1977). Thus involved, the quantity of information can be interpreted as the quantity of “substance” or “being” of that structure allowing of self-grounding.
Skolem’s “Paradox” as Logic of Ground: the Mutual Foundation of Both Proper and Improper Interpretations