[Paper Review] Inconsistency of Primitive Recursive Arithmetic

Edward Nelson|arXiv (Cornell University)|Sep 30, 2015

Mathematical and Theoretical Analysis1 references3 citations

TL;DR

This paper presents a formal proof of the inconsistency of primitive recursive arithmetic (PRA) using Nelson's qea proof system, a computational framework that validates and expands mathematical proofs via a Perl-based checker. The proof demonstrates a contradiction in PRA by deriving both a statement and its negation through syntactic manipulation and equality substitutions, ultimately concluding with a formal contradiction labeled 'QEA'.

ABSTRACT

The aim of this work is to show that contemporary mathematics, including Peano arithmetic, is inconsistent, to construct firm foundations for mathematics, and to begin building on these foundations.

Motivation & Objective

• To investigate the consistency of primitive recursive arithmetic (PRA) using a formal proof system.
• To verify the correctness of a proof of theorem 13i in Nelson's Balrog system through automated expansion.
• To demonstrate a contradiction in PRA by deriving both a statement and its negation through equality substitutions and inference rules.
• To validate the logical structure of the proof using the qea system, which checks correctness and generates expanded proof forms.
• To contribute to foundational debates on the consistency of arithmetic by exposing a formal inconsistency in PRA.

Proposed method

• The qea proof system parses TeX-encoded proof sources and validates each step using predefined inference rules.
• Special cases and previously established results (e.g., 0+x = x+0, x+0 = x) are used as premises in the proof.
• Equality substitutions are applied to derive new statements by replacing terms in equations, with redacted text indicating substitution points.
• Inference steps are chained using numbered references, where each new line depends on prior lines and logical rules.
• The proof uses syntactic manipulation to derive both a statement and its negation, leading to a contradiction.
• The final line 'QEA' signals the detection of a contradiction, confirming inconsistency.

Experimental results

Research questions

• RQ1 Can the qea proof system detect inconsistencies in primitive recursive arithmetic?
• RQ2 Does the proof of theorem 13i in Nelson's Balrog system lead to a formal contradiction?
• RQ3 Can equality substitutions and inference rules in PRA generate both a statement and its negation?
• RQ4 Is the claim of inconsistency in PRA logically valid under the qea system's verification?
• RQ5 What role does automated proof expansion play in exposing logical flaws in foundational systems?

Key findings

• The proof derives both 'S'(x+0) = 0+'S'x and its negation, leading to a contradiction.
• The system identifies a contradiction through the derivation of ¬'S'(x+0) = 0+'S'x and 'S'(x+0) = 0+'S'x.
• The contradiction arises from the combination of equality substitutions and inference steps on the basis of prior axioms and lemmas.
• The qea system confirms the inconsistency by labeling the final contradiction as 'QEA'.
• The proof demonstrates that assuming the theorem's negation leads to a logical inconsistency in PRA.
• The expanded proof reveals that the system's inference rules generate mutually exclusive statements, invalidating the consistency of PRA.