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 recursively enumerable (r.e.) set of axioms. Gödel provided a negative answer, stating that no such extension exists.

The article argues that the absence of algorithmic solutions does not necessarily imply the impossibility of completing PA. It presents a realistic means of completing PA by allowing a random choice of one pair of axioms in each pair, which assures such completion with a probability of 99%. This cannot be done for PA itself, and the article further discusses the implications of this impossibility.

The article also introduces the concept of mutual information in two finite strings, which can be refined and extended to infinitesequences. It states that no physically realizable process can increase information about a specific sequence, and this leads to a negative answer to the question of whether non-mechanical means could enable the Hilbert-Gödel task of consistent completion for PA.

Link to Article: https://arxiv.org/abs/0203029v18 Authors: arXiv ID: 0203029v18