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An undecidable extension of Morley's theorem on the number of countable models |
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| Institution: | 1. University of Victoria, Department of Mathematics and Statistics, PO BOX 1700 STN CSC, Victoria, British Columbia, V8W 2Y2, Canada;2. University of Toronto, Department of Mathematics, 40 St. George St., Toronto, Ontario, M5S 2E4, Canada;3. Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria |
| Abstract: | We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory. |
| Keywords: | Morley's theorem Countable models Random and Cohen forcing Woodin cardinals Inner model theory |
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