arXiv subjects
Publications and source records attributed to Michael Reeken.
We study the structure of effective cardinals in the nonstandard set universe of Hrbacek set theory HST. Some results resemble those known in descriptive set theory in the domain of Borel reducibility of equivalence relations.
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed under the Fubini product modulo the ideal Fin of all finite sets, and some other operations. Our second main result shows that T_2, the equivalence relation called ``the equality of countable sets of the reals'', is not Borel reducible to another family of ERs which includes continuous actions of Polish CLI groups, Borel equivalence relations with G_δclasses, some ideals, and is closed under the Fubini product over $Fin.$ Both results and their corollaries extend some earlier irreducibility theorems by Hjorth and Kechris.
We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to those discovered in descriptive set theory. The Borel reducibility structure of Borel sets and (partially) CD reducibility structure of CD sets in *N is described. We prove that all CD ERs with countable equivalence classes are CD-smooth, but not all are B-smooth, for instance, the ER of having finite difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces, any CD ER on *N either has at most continuum-many classes or there is an infinite internal set of pairwise inequivalent elements. Our study of monadic ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive Borel cut in *N, shows that these ERs split in two linearly families, associated with countably cofinal and countably coinitial cuts, each of which is linearly ordered by Borel reducibility. The relationship between monadic ERs and the ER of finite symmetric difference on hyperfinite subsets of *N is studied.
We characterize those standard models M of ZFC which are embeddable, as the class of all standard sets, in a model of internal set theory IST. The necessary and sufficient condition is that 1) there is a wellordering < of M which does not destroy the ZFC schemata, and 2) the truth relation for (M,<) does not destroy Separation. The result is interpreted as the answer for the question in the title.
A new elementary nonstandard proof of the Jordan curve theorem is given. The proof (the technical part consists of 4 pages) is self-contained, except for the Jordan theorem for polygons taken for granted.
We study models of HST, a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets. This theory admits an adequate formulation of the isomorphism property IP: "any two elementarily equivalent internally presented structures of a wellorderable language are isomorphic." IP implies, for instance, that all infinite internal sets are equinumerous, and there exists a unique (modulo isomorphisms) internal elementary extension of the standard reals. We prove that IP is independent of HST (using the class of all sets constructible from internal sets) and consistent with HST (using generic extensions of $\HST$ models by a sufficient number of generic isomorphisms).