Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778151 | Annals of Pure and Applied Logic | 2017 | 23 Pages |
Abstract
Let FÏ1 be the countable admissible ordinal equivalence relation defined on 2Ï by xFÏ1y if and only if Ï1x=Ï1y. Some invariant descriptive set theoretic properties of FÏ1 will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed FÏ1 is not the orbit equivalence relation of a continuous action of a Polish group on 2Ï. Becker stengthened this to show FÏ1 is not even the orbit equivalence relation of a Î11 action of a Polish group. However, Montalbán has shown that FÏ1 is Î11 reducible to an orbit equivalence relation of a Polish group action, in fact, FÏ1 is classifiable by countable structures. It will be shown here that FÏ1 must be classified by structures of high Scott rank. Let EÏ1 denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, Eâ¤aÎ11F denotes the existence of a Î11 function f:XâY which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies EÏ1â¤aÎ11FÏ1. However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), EÏ1â¤aÎ11FÏ1 is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Î11 reducible to FÏ1. This shows the consistency of a negative answer to a question of Sy-David Friedman.
Related Topics
Physical Sciences and Engineering
Mathematics
Logic
Authors
William Chan,