Dedekind-finite cardinals and model-theoretic structures

The notion of finiteness in the absence of AC has been widely studied. We consider a minimal criterion for which any class of cardinalities that satisfies it can be considered as a finiteness class. Fourteen notions of finiteness will be presented and studied in this thesis. We show how these classes relate to each other, and discuss their closure properties. Some results can be proved in ZF. Others are consistency results that can be shown by using the Fraenkel-Mostowski-model construction. Furthermore we investigate the relationship between Dedekind-finite sets and definability, and try to carry out reconstruction to recover the original structures used to construct FM-models. Later we establish a connection between tree structures and sets with their cardinalities in one of the finiteness classes, written as ∆₅.