Robinson Splitting Theorem and $\Sigma_1$ Induction

TL;DR

This paper explores a variant of the Robinson Splitting Theorem within models of weak arithmetic, showing that a weaker form holds when replacing lowness with superlowness in certain logical systems.

Contribution

It extends the Robinson Splitting Theorem to models of ext{P}^-+ ext{I} ext{Σ}_1, demonstrating that a weaker splitting property holds under superlowness instead of lowness.

Findings

A weaker splitting theorem holds in models of P^-+IΣ_1.

Superlowness replaces lowness in the splitting context.

The result connects computability theory with models of weak arithmetic.

Abstract

The Robinson Splitting Theorem states that a c.e. degree $\mathbf{b}$ splits over any low c.e. degree $\mathbf{c}<\mathbf{b}$. We prove that a weaker version of this theorem holds in models of $\mathrm{P}^-+\mathrm{I}\Sigma_1$, with lowness replaced by superlowness.