The monadic theory and the “next world” |
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| Authors: | Yuri Gurevich Saharon Shelah |
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| Institution: | (1) Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel;(2) Present address: Department of Computer Science, The University of Michigan, 48109 Ann Arbor, MI, USA;(3) Present address: Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel |
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| Abstract: | Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderV
B-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity
of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the
real line. Letr be a Cohen real overV. The second-orderVr]-theory of ℵ0 is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderVr]-theory of the real line is interpretable in the monadicV-theory of the real line.
Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death
The author thanks the United States-Israel Binational Science Foundation for supporting the research. |
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