How strong are single fixed points of normal functions?

Abstract

In a recent paper by M. Rathjen and the present author it has been shown that the statement ``every normal function has a derivative'' is equivalent to ${\Pi }_1^1$-bar induction. The equivalence was proved over $\mathbf{AC}{\mathbf{A}}_0$, for a suitable representation of normal functions in terms of dilators. In the present paper we show that the statement ``every normal function has at least one fixed point'' is equivalent to ${\Pi }_1^1$-induction along the natural numbers.

References in corpus (5)

• ${\Pi }_1^1$-Comprehension as a Well-Ordering Principle
• Computable Aspects of the Bachmann-Howard Principle
• Derivatives of normal functions in reverse mathematics
• A Categorical Construction of Bachmann-Howard Fixed Points
• A note on ordinal exponentiation and derivatives of normal functions

Cited by in corpus (1)

• Derivatives of normal functions in reverse mathematics