[Paper Review] Is mathematics consistent?

Hitoshi Kitada|ArXiv.org|May 31, 2003

Mathematics Education and Teaching Techniques3 references3 citations

TL;DR

This paper investigates the consistency of formal mathematical systems, particularly set theory, by extending Gödel's incompleteness theorems via Rosser's refinement. It shows that if ZFC's axioms cannot determine the existence of nonrecursive ordinals, then either all ordinals are recursive (implying set theory is inconsistent) or there exists a least nonrecursive ordinal, leading to a maximal consistent extension of the system. The key contribution is a metamathematical analysis linking ordinal recursiveness to foundational consistency.

ABSTRACT

A question is proposed whether or not set theory is consistent.

Motivation & Objective

• To assess the consistency of formal set theory S^(0) using Gödel numbering and proof predicates.
• To extend Gödel's incompleteness theorem using Rosser's form to avoid the standard assumption of ω-consistency.
• To analyze the implications of the independence of nonrecursive ordinal existence from ZFC axioms on the consistency of extended formal systems.
• To determine whether the existence of a least nonrecursive ordinal leads to a maximal consistent extension of a formal system.
• To clarify whether the apparent inconsistency in case (i) implies a concrete contradiction within ZFC or is a metamathematical limitation.

Proposed method

• Uses Gödel numbering to assign unique numbers to formulas and proofs in a formal system S^(0), enabling arithmetization of syntax.
• Defines primitive recursive predicates A^(0)(a,b) and B^(0)(a,c) representing provability and refutability of formulas with Gödel number a and proof number b or c.
• Constructs a self-referential formula A_q^(0)(q^(0)) analogous to the Gödel sentence, using diagonalization to create a formula asserting its own unprovability.
• Applies Rosser’s trick by considering both provability and refutability in the same formula, avoiding the need for ω-consistency.
• Extends the system S^(0) to S^(1) by adding A_(0) as a new axiom, preserving consistency under the assumption that neither A_q^(0)(q^(0)) nor its negation is provable.
• Analyzes the transfinite extension of the system S^(α) through countable ordinals, distinguishing between cases where all ordinals are recursive or a least nonrecursive ordinal ω_1 exists.

Experimental results

Research questions

• RQ1 Under what conditions is a formal system S^(0) consistent when extended via self-referential formulas constructed via Gödel numbering?
• RQ2 Can Rosser’s form of the incompleteness theorem be used to establish the unprovability of a sentence asserting its own unprovability without assuming ω-consistency?
• RQ3 What happens to the consistency of a formal system when extended through transfinite ordinals, particularly when the existence of nonrecursive ordinals is independent of ZFC?
• RQ4 If all ordinals are recursive, does this lead to a contradiction in the consistency of transfinite extensions of formal systems?
• RQ5 Is the existence of a least nonrecursive ordinal sufficient to define a maximal consistent extension of a formal system S^(0)?

Key findings

• The system S^(0) is consistent if and only if neither A_q^(0)(q^(0)) nor its negation is provable, as shown via Rosser’s refinement of Gödel’s theorem.
• If all ordinals are recursive, then the transfinite extension of S^(α) must eventually fail to remain consistent, leading to a contradiction unless set theory is inconsistent.
• The existence of a least nonrecursive ordinal ω_1 ensures that the system S^(ω_1) is consistent and cannot be extended further without violating consistency.
• The system S^(β) reaches a maximal consistent state precisely at β = ω_1, the least nonrecursive ordinal, if such an ordinal exists.
• The theorem is metamathematical: it does not prove inconsistency within ZFC, but shows that inconsistency arises only if all ordinals are recursive and the system is assumed to be indefinitely extendable.
• Even if no nonrecursive ordinal exists, the resulting contradiction does not yield a concrete inconsistency (like Russell’s paradox) within ZFC, but rather a structural limitation in the extension process.