Leo Harrington showed that the second-order theory of arithmetic WKL0 is ${\Pi^1_1}$-conservative over the theory RCA0. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.

Proof Theory

Harrington’s conservation theorem redone

An inferential semantics for full Higher Order Logic (HOL) is proposed. The paper presents a constructive notion of model, that being able to capture relevant computational aspects is particularly suited for the applications of HOL to computer science. The inferential semantics is based on the introduction of new abstract deduction structures (ADS) that express the action of the Comprehension Axiom in a Higher Order Logic proof. The ADS's allow to define an inferential algebra of higher order potential proof-trees, endowed with two binary operations, the abstraction and the contraction, each consisting of constructive reductions between potential proofs. Typed formulas are interpreted by sequent trees, and the operations between trees correspond to the logical connectives of the interpreted formula. Higher order logic is sound and complete w.r.t. the given inferential semantics.

Abstract deduction and inferential models for type theory

The original Goodstein process proceeds by writing natural numbers in nested exponential k-normal form, then successively raising the base to $k+1$ and subtracting one from the end result Such sequences always reach zero, but this fact is unprovable in Peano arithmetic In this paper we instead consider notations for natural numbers based on the Ackermann function We define two new Goodstein processes, obtaining new independence results for $\mathsf{ACA}_0'$ and $\mathsf{ACA}_0^+$, theories of second order arithmetic related to the existence of Turing jumps

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Ackermannian Goodstein sequences of intermediate growth

This paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

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About the Proof-Theoretic Ordinals of Weak Fixed Point Theories

Michael Rathjen and the present author have shown that $\Pi^1_1$-bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in $\mathbf{ACA_0}$. In this note we show that the base theory can be weakened to $\mathbf{RCA_0}$. Our argument makes crucial use of a normal function $f$ with $f(\alpha)\leq 1+\alpha^2$ and $f'(\alpha)=\omega^{\omega^\alpha}$. We will also exhibit a normal function $g$ with $g(\alpha)\leq 1+\alpha\cdot 2$ and $g'(\alpha)=\omega^{1+\alpha}$.

Proof Theory

Citations

Harrington’s conservation theorem redone

Abstract deduction and inferential models for type theory

Ackermannian Goodstein sequences of intermediate growth

About the Proof-Theoretic Ordinals of Weak Fixed Point Theories

A note on ordinal exponentiation and derivatives of normal functions

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