Relating some small ordinal naturally to commutative algebra

Andreas Weiermann, University of Ghent. Part of the logic seminar series.

Let $R=F[x_1,ldots,x_n]$ be the ring of polynomials in $n$ variables over a field $F$. By Hilbert\'s basis theorem every increasing infinite sequence of ideals in R stabilises. The resulting ordinal height of the reverse inclusion relation on ideals is equal to $\omega^k$. Quite interestingly, already for $n=2$ the maximal linear extension of the reverse inclusion relation (which exists by MacLagan\'s theorem) has order type $\omega^{\omega+2}$ (Altman, unpublished).

We will indicate a proof of this by using ordinals related to finite and infinite Young diagrams and we will show that resulting lengths of effectively listed bad sequences of ideals is bounded from below by a two-fold iteration of the Ackermann function.If time remains we will also discuss the case $n>2.$

Andreas Weiermann, University of Ghent