# Better quasi-orders and iterated ideals

## Giovanni Solda

**Authors:** Giovanni Solda and Fedor Pakhomov

The notion of better quasi-order (henceforth bqo) is a natural evolution of that of well quasi-order (henceforth wqo). Intuitively, a wqo is bqo if all of its iterated powersets are also wqo’s: this gives bqo’s much stronger closure properties than the ones wqo’s enjoy.

In this talk, we give a new characterization of bqo’s.
Inspired by the intuition we gave above, we look
at the class of **iterated ideals** of a wqo:
this produces a much smaller object than the
iterated powerset.
Our main result is that this smaller object still
contains a lot of information on the original wqo:
namely, a wqo \(Q\) is bqo if and
only if the iterated ideals of \(Q\) form a wqo, and
the formalization of this result in second-order
arithmetic is provable over \(\mathsf{ATR}_0\). We will then show
some properties of the class of iterated ideals,
and point at some (potential) applications.