Reflexivity and the diagonal argument in proofs of limitative theorems

TL;DR

This paper critically examines the limitations of reflexive and diagonal arguments in proving limitative theorems, proposing an algorithm that challenges traditional conclusions about computability and formal systems.

Contribution

It introduces an algorithm for generating real numbers that questions the assumptions of existing limitative theorems and explores their implications for theoretical computer science.

Findings

Reflexivity in formal systems does not preclude solutions within the same system.

Diagonal arguments do not necessarily prove the non-existence of a universal real number generator.

Proposes an algorithm capable of generating all real numbers, challenging classical limitative theorems.

Abstract

This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal system contains a sentence, which introduces reflexitivity, does not imply, that the same system does not contain a sentence or a proof procedure which solves this problem. Second basic method of proof - diagonal argument (i.e. showing non-eqiunumerosity of a program set with the set of real numbers) does not exclude existance of a single program, capable of computing all real numbers. In this work, we suggest an algorithm generating real numbers (arbitrary, infinite in the limit, binary strings), and we speculate it's meaning for theoretical computer science.