2,671 research outputs found
A partition theorem for ordinals
AbstractIn this note we show that Ī± ātrans(Ī±, m)2 for all m < Ļ and every Ī± of the form ĻĪ², where Ī± ātrans(Ī±, m)2 is a weakening of the usual partition property obtained by considering only partitions whose first member is a transitive relation
The topological pigeonhole principle for ordinals
Given a cardinal and a sequence
of ordinals, we determine the least ordinal (when one exists) such that
the topological partition relation
holds, including
an independence result for one class of cases. Here the prefix "" means
that the homogeneous set must have the correct topology rather than the correct
order type. The answer is linked to the non-topological pigeonhole principle of
Milner and Rado.Comment: 24 page
The bounded proper forcing axiom and well orderings of the reals
We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(Ļ_1) which is Ī_1 definable with parameter a subset of Ļ_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of Ļ_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the HƤrtig quantifier is not lightface projective
Some natural zero one laws for ordinals below Īµ0
We are going to prove that every ordinal Ī± with Īµ_0ā>āĪ±āā„āĻ^Ļ satisfies a natural zero one law in the following sense. For Ī±ā<āĪµ_0 let NĪ± be the number of occurences of Ļ in the Cantor normal form of Ī±. (NĪ± is then the number of edges in the unordered tree which can canonically be associated with Ī±.) We prove that for any Ī± with Ļ Ļ āā¤āĪ±ā<āĪµ_0 and any sentence Ļ in the language of linear orders the asymptotic density of Ļ along Ī± is an element of ā{0,1}. We further show that for any such sentence Ļ the asymptotic density along Īµ_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below Ļ^Ļ
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