A note on ordinal exponentiation and derivatives of normal functions

Abstract

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 $\mathbf{AC}{\mathbf{A}}_0$. In this note we show that the base theory can be weakened to $\mathbf{RC}{\mathbf{A}}_0$. Our argument makes crucial use of a normal function $f$ with $f(\alpha )\le 1+{\alpha }^2$ and $f^{\prime }(\alpha )={\omega }^{{\omega }^{\alpha }}$. We will also exhibit a normal function $g$ with $g(\alpha )\le 1+\alpha \cdot 2$ and $g^{\prime }(\alpha )={\omega }^{1+\alpha }$.

References in corpus (2)

• Computable Aspects of the Bachmann-Howard Principle
• Derivatives of normal functions in reverse mathematics

Cited by in corpus (2)

• Derivatives of normal functions in reverse mathematics
• How strong are single fixed points of normal functions?