The Impossibility of Consistent Extension for Peano Arithmetic

Title: The Impossibility of Consistent Extension for Peano Arithmetic

Abstract: This research article explores the possibility of extending Peano Arithmetic (PA) to a complete theory. PA is a system of mathematics that consists of logical and algebraic axioms, along with an infinite family of Induction Axioms. The main question posed by David Hilbert was whether it is possible to extend PA to a complete theory, particularly focusing on recursively enumerable (r.e.) extensions.

Kolmogorov complexity, a concept introduced by Andrei Kolmogorov, plays a crucial role in this research. It measures the minimum length of a computer program that can generate a given string, assuming the program and the string are both encoded in the same way. The article argues that any extension of PA that leaves an n-bit input unresolved or contains nearly all information about an n-bit prefix of any r.e.real sequence is impossible.

The article also discusses the concept of mutual information in two finite strings, which can be refined and extended to infinitesequences. It suggests that no physically realizable process can increase information about a specific sequence, implying that non-mechanical means cannot enable the consistent completion of PA.

In conclusion, the research article provides a negative answer to Hilbert's question, suggesting that it is impossible to extend PA to a complete theory using any realistic means.

Link to Article: https://arxiv.org/abs/0203029v20 Authors: arXiv ID: 0203029v20