Fleeting thoughts: Groundability of mathematics by Hilbert’s ε-symbol

Prehistory and background: 

Hilbert (also in collaboration with Ackermann and Bernays) introduced the concept of ε-symbol for the self-foundation of mathematics in the framework of his formalist program in a few classical and well-known papers. The essence of ε-symbol consists in the disjunctive distinction of the “bad” pure existence in mathematics from the “good” one. The former was alleged as the source of all paradoxes in set theory while the latter was able to conserve the power of mathematics involving (actual) infinity in a consistent way. The “ε-symbol” defines right the latter case adding to the quantifier of existence the condition of “if at least one exists” therefore excluding the cases of existence by an empty set in the “pure existence”. Both quantifiers ∃,∀ are expressible by ε. The ε-symbol represents a form of the axiom of choice where the degenerated cases of choices of elements of and from an empty set are not allowed. 

Roughly speaking, the “ε-symbol” can ground all mathematics, but not itself. So, Hilbert’s program of formalism needs crucially the self-foundation of the “ε-symbol” to be accomplished.

The intention is a probabilistic interpretation of “ε-symbol”: thus, a generalized arithmetic based on the complex Hilbert space to ground itself and “ε-symbol”; therefore to finish Hilbert’s program.

Thesis: 

The “ε-symbol” can be interpreted as a certain probability distribution as to any infinite set. Any probability distribution by its corresponding characteristic function is an element of the complex Hilbert space. The complex Hilbert space is a generalization of Peano arithmetic where the natural number n is generalized to the nth qubit of the complex Hilbert space. The relation of Peano arithmetic, being contained as a true structure in the complex Hilbert space, and any qubit defines both transfinite induction and the “ε-symbol” even as equivalent. Thus, Gentzen’s proof (1936, 1938) can serve not less as the self-foundation of “ε-symbol” by the mediation and in the framework of the complex Hilbert space as a generalized Peano arithmetic. 

A few comments of the thesis:

The axiom of choice together with Peano arithmetic implies the relativeness of ‘set’ known also as Skolem’s paradox (1922). Set theory (e.g. ZFC) postulates infinite sets and does not include the axiom of induction of Peano arithmetic. The axiom of choice generates some well-ordering (Zermelo 1904) equivalent to an initial segment of natural numbers for any infinite set. All natural numbers are finite (1 is finite; adding 1 to any finite number, a finite number is obtained; consequently, all natural numbers are finite according to the axiom of induction). Thus, any infinite set corresponds to some finite set under the above conditions. This can satisfy the “ε-symbol” if that finite set is always randomly chosen and thus generates a probability distribution for all initial segments of natural numbers. That probability distribution defines unambiguously an infinite set, and all sets whether finite or infinite are unambiguously represented in the complex Hilbert space.