[Paper Review] On the arithmetization of syntax

Stephen Boyce|arXiv (Cornell University)|Oct 6, 2010

Computability, Logic, AI Algorithms1 references3 citations

TL;DR

This paper introduces a modified first-order arithmetic system S′ that parameterizes individual variables to range only over objects assigned to numerals under any interpretation. By adapting Gödel's technique, it constructs a Gödel sentence (∀x)ℛ(x) and proves S′ inconsistent, implying that standard first-order arithmetic S is also inconsistent, challenging foundational assumptions of formalism in arithmetic.

ABSTRACT

It is generally accepted that Godel's proof implies the incompleteness of first-order number theory. This paper shows that the standard demonstration of this result implies that the class of theorems of such systems is not well defined. Proof of the converse of one of the Hilbert-Bernays Derivability Conditions for Godel's second incompleteness theorem essentially yields this result: the theoremhood of a formula is implied if it may be proven in the system that there exists a Godel number of a proof of the formula.

Motivation & Objective

• To investigate whether modifying first-order arithmetic to parameterize variables leads to logical inconsistency.
• To assess the implications of such a modification for the consistency of standard first-order arithmetic (S).
• To test the robustness of formalist foundations of arithmetic by exposing potential paradoxes in the metatheory.
• To demonstrate that the standard interpretation of arithmetic cannot serve as a model if the system is inconsistent.
• To challenge the adequacy of formalist methods in avoiding paradox in foundational systems of arithmetic.

Proposed method

• Modify Mendelson’s first-order number theory S by redefining the function s′* to parameterize individual variables so they range only over objects assigned to numerals.
• Preserve all logical axioms, inference rules, and proper axioms of S, ensuring S′ is recursively axiomatized and contains Peano arithmetic.
• Construct a Gödel sentence (∀x)ℛ(x) using a modified version of Gödel’s arithmetization technique, relying on self-reference via substitution and proof predicates.
• Use the completeness of S′ to show that if (∀x)ℛ(x) is not a theorem, then there exists a model where it is false, leading to a contradiction.
• Leverage the fact that for every numeral ḡn, S′ proves ℛ(ḡn), and due to parameterization, this implies (∀x)ℛ(x) must be true in all models.
• Derive a contradiction: if S′ is consistent, then (∀x)ℛ(x) is true in all models and thus a theorem, but this contradicts the construction of the sentence as unprovable.

Experimental results

Research questions

• RQ1 Can a parameterized variable system in first-order arithmetic lead to inconsistency despite preserving standard axioms and rules?
• RQ2 Does the existence of a Gödel sentence in S′ that is both unprovable and necessarily true imply inconsistency in the system?
• RQ3 Is the standard interpretation of first-order arithmetic invalid if S′ is inconsistent due to parameterization?
• RQ4 Can the metatheory of formalist arithmetic be free from paradox if it leads to contradiction under minimal modifications?
• RQ5 What does the inconsistency of S′ imply about the adequacy of formalist accounts of classical mathematics?

Key findings

• S′ is inconsistent because the Gödel sentence (∀x)ℛ(x) must be a theorem due to the parameterization of variables, which forces all instances ℛ(ḡn) to be true in all models.
• The construction leads to a contradiction: if S′ is consistent, then (∀x)ℛ(x) is true in all models and hence a theorem, but this contradicts the assumption that it is unprovable.
• Since the syntax of S′ is identical to that of standard first-order arithmetic S, the inconsistency of S′ implies the inconsistency of S.
• The standard interpretation of S cannot be a model of S′, as it would require (∀x)ℛ(x) to be both true and unprovable, which is impossible under the system’s parameterization.
• The result suggests that the metatheory of formalist arithmetic is not free from paradox, undermining the claim that formalism avoids contradiction in foundational systems.
• The paper concludes that formalist foundations of arithmetic fail to provide a consistent and adequate account of classical mathematics due to hidden paradoxes in the metatheory.