[Paper Review] Comparing WO$(\omega^\omega)$ with $\Sigma^0_2$ induction

Stephen G. Simpson|arXiv (Cornell University)|Aug 11, 2015

Computability, Logic, AI Algorithms5 references3 citations

TL;DR

This paper establishes the logical relationships between WO$(\omega^\omega)$—the statement that the ordinal $\omega^\omega$ is well-ordered—and the subsystems $\Sigma^0_2$ induction (I$\Sigma^0_2$) and $\Sigma^0_2$ bounding (B$\Sigma^0_2$) in second-order arithmetic. It proves that I$\Sigma^0_2$ implies WO$(\omega^\omega)$, but WO$(\omega^\omega)$ does not imply B$\Sigma^0_2$, and their combination does not imply I$\Sigma^0_2$, placing WO$(\omega^\omega)$ strictly between B$\Sigma^0_2$ and I$\Sigma^0_2$ in proof-theoretic strength.

ABSTRACT

Let WO$(\\omega^\\omega)$ be the statement that the ordinal number $\\omega^\\omega$ is well ordered. WO$(\\omega^\\omega)$ has occurred several times in the reverse-mathematical literature. The purpose of this expository note is to discuss the place of WO$(\\omega^\\omega)$ within the standard hierarchy of subsystems of second-order arithmetic. We prove that WO$(\\omega^\\omega)$ is implied by I$\\Sigma^0_2$ and independent of B$\\Sigma^0_2$. We also prove that WO$(\\omega^\\omega)$ and B$\\Sigma^0_2$ together do not imply I$\\Sigma^0_2$.

Motivation & Objective

• To clarify the proof-theoretic strength of WO$(\omega^\omega)$, the statement that $\omega^\omega$ is well-ordered, within the hierarchy of subsystems of second-order arithmetic.
• To determine whether WO$(\omega^\omega)$ implies or is implied by $\Sigma^0_2$ induction (I$\Sigma^0_2$) or $\Sigma^0_2$ bounding (B$\Sigma^0_2$).
• To establish the independence of WO$(\omega^\omega)$ and B$\Sigma^0_2$ over RCA$_0$, and to show that their conjunction does not imply I$\Sigma^0_2$.
• To provide a model-theoretic and proof-theoretic analysis of the logical relationships between WO$(\omega^\omega)$, I$\Sigma^0_2$, and B$\Sigma^0_2$ in the context of reverse mathematics.

Proposed method

• Using $\Pi^0_2$ induction on the natural number $n$ to prove that for any descending sequence $f$ in $\omega^\omega$, the formula $\Phi(n,f)$ holds, which ultimately leads to a contradiction if such a sequence exists.
• Applying $\Pi^0_2$ induction to analyze the structure of descending sequences in $\omega^\omega$, leveraging the Cantor normal form and properties of ordinal arithmetic.
• Constructing a nonstandard model $M$ of B$\Sigma^0_2$ + $\Psi$, then forming an initial segment $\widehat{M}_2$ via $\Sigma^0_2$-definability from a nonstandard element $c$, to analyze the closure properties of the model.
• Using underspill and cofinality arguments to show that $\widehat{M}_2$ satisfies B$\Sigma^0_2$ and WO$(\omega^\omega)$ but not I$\Sigma^0_2$, by demonstrating the existence of a $\Sigma^0_2$-definable surjection from a bounded subset onto an unbounded subset.
• Applying the method of $\Sigma^0_2$-elementary submodels and uniformizing formulas to show that $\widehat{M}_2$ fails to satisfy I$\Sigma^0_2$ while preserving B$\Sigma^0_2$ and WO$(\omega^\omega)$.
• Proving that the consistency of RCA$_0$ + B$\Sigma^0_2$ + WO$(\alpha)$ for a primitive recursive linear order $\alpha$ implies that it does not prove I$\Sigma^0_2$, via a generalization of the model-theoretic construction.

Experimental results

Research questions

• RQ1 Does I$\Sigma^0_2$ imply WO$(\omega^\omega)$ in the context of second-order arithmetic?
• RQ2 Is WO$(\omega^\omega)$ provable in RCA$_0$ + B$\Sigma^0_2$?
• RQ3 Can WO$(\omega^\omega)$ and B$\Sigma^0_2$ together prove I$\Sigma^0_2$?
• RQ4 What is the precise proof-theoretic strength of WO$(\omega^\omega)$ relative to standard subsystems of second-order arithmetic?
• RQ5 Is there a model of RCA$_0$ + B$\Sigma^0_2$ + WO$(\omega^\omega)$ that fails to satisfy I$\Sigma^0_2$?

Key findings

• WO$(\omega^\omega)$ is provable in RCA$_0$ + I$\Sigma^0_2$, as shown via $\Pi^0_2$ induction on the structure of descending sequences in $\omega^\omega$.
• WO$(\omega^\omega)$ does not imply B$\Sigma^0_2$ over RCA$_0$, since the totality of the Ackermann function is provable in RCA$_0$ + WO$(\omega^\omega)$ but not in RCA$_0$ + B$\Sigma^0_2$.
• WO$(\omega^\omega)$ and B$\Sigma^0_2$ together do not imply I$\Sigma^0_2$, as demonstrated by constructing a model $\widehat{M}_2$ satisfying both but not I$\Sigma^0_2$.
• The model $\widehat{M}_2$ is constructed as an initial segment of a nonstandard model $M$ of B$\Sigma^0_2$ + $\Psi$, using $\Sigma^0_2$-definability from a nonstandard element, and shown to satisfy WO$(\omega^\omega)$ and B$\Sigma^0_2$ but not I$\Sigma^0_2$.
• The proof relies on underspill and the failure of $\Sigma^0_2$-induction due to a $\Sigma^0_2$-definable surjection from a bounded subset onto an unbounded subset of $\widehat{M}_2$.
• The results generalize to higher levels: for $k \geq 2$, WO$(\omega_k)$ is provable in RCA$_0$ + I$\Sigma^0_k$ but not in RCA$_0$ + B$\Sigma^0_k$, and their combination does not imply I$\Sigma^0_k$.