On series of ordinals and combinatorics

J. Jones, H. Levitz, W. Nichols

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals $\sum\limits_{\beta<\alpha}\beta$ is the number of 2-element subsets of an ${\alpha}$-element set. It is shown here that for any well-ordered set of arbitrary infinite order type ${\alpha}$, $\sum\limits_{\beta<\alpha}\beta$ is the ordinal of the set $M$ of $2$-element subsets where $M$ is ordered in some natural way. This result is then extended to evaluating the ordinal of the set of all $n$-element subsets for each natural number $n \geq 2$.

Moreover series $\sum\limits_{\beta<\alpha}f(\beta)$ are investigated and evaluated, where ${\alpha}$ is a limit ordinal and the function $f$ belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite ${\alpha}$, but the case of finite ${\alpha}$ appears to be quite problematic.