[Paper Review] The Provability of Consistency

Sergei Artëmov|arXiv (Cornell University)|Feb 20, 2019

Logic, Reasoning, and Knowledge18 references14 citations

TL;DR

This paper presents a formal proof of Peano Arithmetic (PA) consistency within PA itself, bypassing Gödel’s Second Incompleteness Theorem by avoiding the standard arithmetical formula ${\sf Con}_{\sf PA}$. Instead, it formalizes consistency as a scheme using a primitive recursive selector function that verifies, for each finite PA-derivation, that it does not derive contradiction, thus demonstrating that consistency can be proven in PA without internalizing the property as a single formula.

ABSTRACT

We offer a mathematical proof of consistency for Peano Arithmetic PA formalizable in PA. This result is compatible with Goedel's Second Incompleteness Theorem since our consistency proof does not rely on the representation of consistency as a specific arithmetical formula. Our findings show that formal theories can finitely formalize proofs of certain properties presented as schemes without reducing the presentation of those properties to a single formula. We outline a theory of proving schemes in PA.

Motivation & Objective

• To challenge the widespread belief that PA cannot prove its own consistency, as implied by Gödel’s Second Incompleteness Theorem.
• To show that the impossibility reading of Gödel’s theorem is unwarranted when consistency is not reduced to a single arithmetical formula.
• To formalize contentual mathematical reasoning—especially proofs of properties expressed as schemes—within PA, extending its proof-theoretic reach.
• To vindicate Hilbert’s program by demonstrating that consistency proofs can be formalizable in the system they concern, provided schemes are admitted as valid proof forms.

Proposed method

• Formalizing consistency not as a single formula ${\sf Con}_{\sf PA}$, but as a scheme: for any finite PA-derivation $S$, $S$ does not contain $\bot$.
• Introducing an arithmetically definable invariant ${\cal I}_S(\varphi)$ for each formula $\varphi$ in a derivation $S$, such that ${\cal I}_S(\varphi)$ holds for all $\varphi$ in $S$, but ${\cal I}_S(\bot)$ does not hold.
• Using a primitive recursive selector function $t(x)$ that, for each standard numeral $n$, returns a proof $t(n)$ of $\neg n:\bot$, thereby verifying that no standard $n$ proves contradiction.
• Establishing that the entire collection of premises $\neg 0:\bot, \neg 1:\bot, \neg 2:\bot, \ldots$ can be proven in PA via a finite scheme, without proving the universal conclusion $\forall x\, \neg x:\bot$.
• Formalizing the proof of the Complete Induction scheme in PA using Gödel numbering and selector functions, showing that such scheme-based proofs are formalizable.
• Rejecting the ${\sf Con}_{\sf PA}$ as Consistency Principle, which assumes all consistency proofs must be internalized as a single formula, and instead advocating for proofs of schemes as legitimate formal methods.

Experimental results

Research questions

• RQ1 Can the consistency of PA be formally proven within PA if the standard internalization of consistency as ${\sf Con}_{\sf PA}$ is avoided?
• RQ2 Is it possible to formalize contentual mathematical proofs that are not reducible to single formulas, such as proofs of schemes, within PA?
• RQ3 Does Gödel’s Second Incompleteness Theorem genuinely block consistency proofs in PA, or is the obstruction due to an artificial restriction on representation?
• RQ4 Can the notion of formal provability be extended beyond sentences to include schemes, and what are the proof-theoretic consequences?
• RQ5 To what extent does the use of selector functions enable finite formalization of infinite collections of statements in PA?

Key findings

• PA can formally prove its own consistency by representing consistency as a scheme rather than as the single formula ${\sf Con}_{\sf PA}$, thus circumventing Gödel’s Second Incompleteness Theorem.
• The proof of consistency is formalizable in PA using a primitive recursive selector function $t(x)$ that, for each standard $n$, produces a proof $t(n)$ of $\neg n:\bot$, ensuring no standard derivation leads to contradiction.
• The property of Complete Induction is formalized as a scheme in PA, and its proof is formalizable in PA, demonstrating that scheme-based proofs are valid within the system.
• The invariant ${\cal I}_S(\varphi)$ is constructed such that it holds for all formulas $\varphi$ in a derivation $S$, but fails for $\bot$, thereby certifying that $S$ is not a derivation of contradiction.
• The paper refutes the ${\sf Con}_{\sf PA}$ as Consistency Principle, showing that not all consistency proofs need to be internalized as a single formula, and that such internalization is not necessary for formal provability.
• By admitting proofs of schemes as formalizable, the paper extends the scope of formal provability in PA, allowing for self-contained verification of properties like $\forall x[t(x) = 0]$ without relying on external consistency assumptions.