On the monadic theory ofω
1 without A.C. |
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Authors: | Ami Litman |
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Institution: | (1) Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel |
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Abstract: | Buchi inLecture Notes in Mathematics, Decidable Theories II (1973) by using A.C. characterized the theoriesMTβ, <] forβ<ω
1 and showed thatMTω
1, <] is decidable. We extend Buchi’s results to a larger class of models of ZF (without A.C.) by proving the following under
ZF only: (1) There is a choice function which chooses a “good” run of an automaton on countable input (Lemma 5.1). It follows
that Buchi’s results cocerning countable ordinals are provable within ZF. (2) Let U.D. be the assertion that there exists
a uniform denumeration ofω
1 (i.e. a functionf: ω
1 → ω
1
ω
such that for everyα<ω
1,f(α) is a function fromω ontoα). We show that U.D. can be stated as a monadic sentence, and thereforeω
1 is characterizable by a sentence. (3) LetF be the filter of the cofinal closed subsets ofω
1. We show that if U.D. holds thenMTω
1, <] is recursive in the first order theory of the boolean algebraP (ω
1)/F. (We can effectively translate each monadic sentence Σ to a boolean sentenceσ such that ω
1, <] ⊨ Σ iffP(ω
1)/F⊨σ). (4) As every complete boolean algebra theory is recursive we have that in every model of ZF+U.D.,MTω
1, <] is recursive. All our proofs are within ZF. Buchi’s work is often referred to. Following Buchi, the main tool is finite
automata. We don’t deal withMTω
1, <] forω
1 which doesn’t satisfy U.D.
The results in this paper appeared in the author’s M.Sc. thesis, which was prepared at the Hebrew University under the supervision
of Professor M. Rabin. |
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Keywords: | |
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