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


Created:	2015-09-01 12:08
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
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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

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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Partition Relation Equiconsistent with ∃κ(o(κ)=κ+)