Speaker
Alakh Dhruv Chopra
Ghent University
Talks at this conference:
Wednesday, 14:00, J431 
Strength of the hyperated finitary powerset operator 
The finitary powerset operator \(\mathsf{P_f}\) maps a quasiorder \(X\) to the collection of its finite subsets ordered by a certain domination quasiordering (also called the Hoare embedding) while preserving the property of being a wellquasiorder. 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 FernándezDuque and Joosten, and can extended to operators on quasiorders using techniques developed by Provenzano. If \(X\) is a wellquasiorder, then so is \(\mathsf{P^\alpha_f}(X)\) for every \(\alpha\). When \(X\) is a wellorder, this assertion is trivially provable. It explodes in reversemathematical 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 wellquasiorders and of \(\mathsf{P_f}\) specifically, the order type of \(\mathsf{P^{\omega^\alpha}_f}\) is shown to have (fixpointfree) Veblenian lower bounds when considering quasiorders of the form \(\beta \oplus 1\) (aka, wellorders with one incomparable element). This is part of an ongoing project with Fedor Pakhomov, Philipp Provenzano, and Giovanni Soldà to study betterquasiorders and their reversemathematical strength, and can be considered a generalization of the \(\mathsf{H_f}\) operator previously studied by Anton Freund. Bibliography
