# Strength of the hyperated finitary powerset operator

## Alakh Dhruv Chopra

The finitary powerset operator \(\mathsf{P_f}\) maps a quasi-order \(X\) to the
collection of its finite subsets ordered by a certain *domination
quasi-ordering* (also called the *Hoare embedding*) while preserving
the property of being a *well-quasi-order*. This is a widely used and
studied operator in mathematics and computer science. This is extended to its
transfinite *hyperation* \(\mathsf{P^\alpha_f}\) for every ordinal
\(\alpha\), a form of iteration that satisfies the compositional property
\(\mathsf{P^{\alpha+\beta}_f}(X) = \mathsf{P^\alpha_f}(\mathsf{P^\beta_f}(X))\).
Hyperations of ordinal functions were introduced by Fernández-Duque and Joosten,
and can extended to operators on quasi-orders using techniques developed by
Provenzano.

If \(X\) is a well-quasi-order, then so is \(\mathsf{P^\alpha_f}(X)\) for every \(\alpha\). When \(X\) is a well-order, this assertion is trivially provable. It explodes in reverse-mathematical strength as soon as the width of \(X\) is at least 2; for example, the statement for \(\mathsf{P^\omega_f}\) already reaches \(\mathsf{ACA^+_0}\). Using techniques from the study of maximal order types of well-quasi-orders and of \(\mathsf{P_f}\) specifically, the order type of \(\mathsf{P^{\omega^\alpha}_f}\) is shown to have (fixpoint-free) Veblen-ian lower bounds when considering quasi-orders of the form \(\beta \oplus 1\) (aka, well-orders with one incomparable element).

This is part of an ongoing project with Fedor Pakhomov, Philipp Provenzano, and Giovanni Soldà to study better-quasi-orders and their reverse-mathematical strength, and can be considered a generalization of the \(\mathsf{H_f}\) operator previously studied by Anton Freund.

## Bibliography

- David Fernández-Duque, Joost J. Joosten,
*Hyperations, Veblen progressions and transfinite iteration of ordinal functions*,, vol. 164, issues 7–8, pp. 785–801.*Annals of Pure and Applied Logic* - Philipp Provenzano,
*The reverse mathematical strength of hyperations*,, 2022,*Master’s Thesis**(Unpublished)*.