Partition Relation Equiconsistent with ∃κ(o(κ)=κ+)
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Description: |
Kimchi, YM (Technion - Israel Institute of Technology)
Wednesday 26 August 2015, 14:00-14:30 |
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Created: | 2015-09-01 12:08 |
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Collection: | Mathematical, Foundational and Computational Aspects of the Higher Infinite |
Publisher: | Isaac Newton Institute |
Copyright: | Kimchi, YM |
Language: | eng (English) |
Distribution: | World (downloadable) |
Explicit content: | No |
Aspect Ratio: | 16:9 |
Screencast: | No |
Bumper: | UCS Default |
Trailer: | UCS Default |
Abstract: | Preamble: In this work we deal with partition relations with infinite exponents under ZFC, hence all results are limited to definable functions.
In [78], M. Spector has proven, basically, that ∃κ(o(κ)=1) is equiconsistent with ℵ1→(ω)ωℵ0. In [87], we were able to show that the result generalizes to n=2; namely, ∃κ(o(κ)=2) is equiconsistent with ℵ1→(ω2)ω2ℵ0. Surprisingly at first sight, this property cannot be generalized further (for n>2), and later on we were able to prove that ℵ1→(ω3)ω3ℵ0 is equiconsistent with ∃κ(o(o(κ))=2). The above lead us to a finer notion of homogeneity: Definition: Weak Homogeneity is the partition property κ−→−−−−\tiny WH(λ)ημ where the only considered subsequences of λ are those that are created by removing (or, complementarily, collecting) only finitely many segments of λ. Using week homogeneity we were able to prove the following for any ordinal α [87]: ∃κ(o(κ)=α) is equiconsistent with ℵ1−→−−−−\tiny WH(ωα)ωαℵ0. Later on we were able to characterize the consistency strength of ∃κ(o(κ) = κ), and recently we have arrived at the main result of this paper: ∃κ(o(κ)=κ+) is equiconsistent with ℵ1−→−−−−\tiny WH(ℵ1)ℵ1ℵ0 References: [78] M. Spector: Natural Sentences of Mathematics which are independent of V=L, V=Lμ etc., 1978 (preprint). [87] Y.M. Kimchi: Dissertation, 1987, The Hebrew University of Jerusalem, Israel |
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