Hilbert Spaces Without Countable AC

This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.Comment: 51 page

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

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