433 research outputs found
Cohesive avoidance and arithmetical sets
An open question in reverse mathematics is whether the cohesive principle,
\COH, is implied by the stable form of Ramsey's theorem for pairs,
\SRT^2_2, in -models of \RCA. One typical way of establishing this
implication would be to show that for every sequence of subsets of
, there is a set that is in such that every
infinite subset of or computes an -cohesive set. In this
article, this is shown to be false, even under far less stringent assumptions:
for all natural numbers and , there is a sequence \vec{R}
= \sequence{R_0,...,R_{n-1}} of subsets of such that for any
partition of arithmetical in , there is an
infinite subset of some that computes no set cohesive for . This
complements a number of previous results in computability theory on the
computational feebleness of infinite sets of numbers with prescribed
combinatorial properties. The proof is a forcing argument using an adaptation
of the method of Seetapun showing that every finite coloring of pairs of
integers has an infinite homogeneous set not computing a given non-computable
set
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