Ordinal collapsing function: Difference between revisions

The first ordinal collapsing function in the literature was Bachmann's \( \psi \) function, which was regarded as novel at the time and was used to calibrate the size of the [[Bachmann-Howard ordinal]]. However, the definition is quite cumbersome. A modern "recast", proposed by Michael Rathjen<ref>Rathjen, Michael. "The Art of Ordinal Analysis"</ref>, is that \( \psi_\Omega(\alpha)\) is the least countable \( \rho \) so that all countable ordinals reachable from ordinals below \( \max(1, \rho) \) and \( \Omega \) via addition, the map \( \xi \mapsto \omega^\xi \), and \( \psi_\Omega \) restricted to inputs less than \( \alpha \) are less than \( \rho \). Of course, this definition is condensed, and is usually written in terms of \( C \)-sets. Below is the more formal definition.