# Skolem’s “Paradox” as Logic of Ground: The Mutual Foundation of Both Proper and Improper Interpretations

### EasyChair Preprint no. 3709

16 pages•Date: July 1, 2020### Abstract

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality.

Its investigation needs philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction.

A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis.

Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom.

**Keyphrases**: axiom of choice, axiom of induction, axiom of transfinite induction, eidetic or phenomenological or transcendental reduction epoché, Gödel mathematics, Hilbert mathematics, Husserl's Phenomenology, information, principle of universal mathematizability, quantum information, quantum mechanics