Set Theory
This page hopes to record some of the progress made by the Cambridge Set Theory group…
The group has been examining the properties of models of the stratified axioms of Zermelo - Fraenkel Set Theory (ZF) produced using techniques developed in .<<http://www.dpmms.cam.ac.uk/~tf/strZF.pdf>> In this paper two models are constructed.. The first, denoted HS, is the collection of hereditarily symmetric sets generated by a group of permutations (Forster specifically chooses the generating group to be a subgroup of the permutations of V_of finite support, but that was a particularly simple example chosen merely to get the ball rolling). The second model, S, is the stratified analogue of L. It is build by closing under rudimentary stratified operations.
Some important definitions and abbreviations are:
strZF- the stratified axioms of ZF IO- the assertion that every set is the same size as a set of singletons SC- the assertion that every set is strongly cantorian
Kaye-Forster Set Theory (KF) consists of the axioms: extensionality, pairing, union, power set and bounded stratified comprehension.
One goal of the group is to find a model of KF that refutes IO. Nathan Bowler has shown that if HS is generated by a group of permutations of finite support that is a set, then HS satisfies IO. This result can be found in . http://www.dpmms.cam.ac.uk/~zjm20/HSSurveyArticleV.pdf <<>>
Why do we want to refute IO? The idea is to find a model of KF in which Hartogs’ theorem fails, and refuting IO would be a start (We would need a wellordered counterexample to IO) Such a model would have a set of all ordinals (in some sense: a set containing wellorderings of all lengths) and it would be a first step on the road to a model of NF.
All known proofs of Hartogs’ theorem use at least some unstratified separation. If we can prove Hartogs’ theorem using stratified separation only then NF will be inconsistent. So can we find a model of KF where unstratified separation fails? Yes, the symmetric models discovered by Forster. Can we find such models where choice holds? This is a question asked by Mac Kenzie. It is a curious circumstance that at the time of writing the only models of KF which violate unstratified separation are the symmetric models.
A question of Mackenzie: We know that for every α there must be a β such that
Jα ∩ S ⊆ Sβ
by a brutal argument using replacement - in just the way we use replacement to show that power set holds in L. Can we give a sensible bound for β in terms of α?
Annoyingly HS contains hidden inside itself a complete description of everything in the Mother Ship. Vu Dang spotted this with his usual sharpness. Let x be any set whatever. Think of x as a tree whose vertices are elements of TCl(x) and where the edges are all ∈. Now replace, in this tree, every endpoint by the tree corresponding to Vω. The result is the ∈-picture of a hereditarily symmetric set. Extremely vexing.
