We can compare the relative sizes of sets by using injections or (partial) surjections, but without the axiom of choice we cannot prove that every two sets can be compared. We can use the ordinals to define a notion of size which allows us to determine whether a set is "large" or "small" relative to another. The first is defined by the Hartogs number, which is the least ordinal which does not inject into the set; the second is the Lindenbaum number of a set, which is the first ordinal which is not an image of the set. In this talk we will discuss some basic properties of these numbers and some basic historical results.
In a new work with Calliope Ryan-Smith we showed that given any pair of (infinite) cardinals, we can onstruct a symmetric extension in which there is a set whose Hartogs is the smaller and the Lindenbaum is the larger. Moreover, using the techniques of iterated symmetric extensions, we can realise all possible pairs in a single model.
This work appears on arXiv: https://arxiv.org/abs/2309.11409