On Unsound Linear Orderings

09.05.2023 15:00 - 16:30

T. Weinert (U Wien)

In the Eighties Adrian Mathias introduced the notion of soundness of an ordinal. An ordinal is sound if for any countable partition \(P\) of it only countably many ordinals are order-types of unions of subpartitionts of \(P\). Mathias showed that the least unsound ordinal \(\zeta\) is \(\omega_1^{\omega + 2}\) if \(\aleph_1\) can be embedded into the continuum but if \(\aleph_1\) is regular yet cannot be embedded into the continuum, \(\zeta \geqslant \omega_1^{\omega 2 + 1}\).

I am going to discuss his findings and consider the notion for the more general class of linear orderings building on work by him, MacPherson, and Schmerl. I am also going to mention some open problems. This is joint ongoing work with Garrett Ervin and Jonathan Schilhan.




SR 10, 1. Stock, Koling. 14-16, 1090 Wien