Conservativity of an ultrafilter of IP sets over \a0ATR0{\mathbf{ATR_0}}

An IP set is a set of natural numbers containing all finite sums of an infinite sequence of natural numbers. Let U\mathfrak{U} be an ultrafilter on N\mathbb{N}, and let ATR0\mathbf{ATR_0} denote the corresponding subsystem of second-order arithmetic. IP-ultrafilter conservativity conjecture. The theory

ATR0+U+“every element of U is IP”\mathbf{ATR_0}+\exists\mathfrak{U}+\text{“every element of $\mathfrak{U}$ is IP”}

is a conservative extension of ATR0\mathbf{ATR_0}. This asks whether the existence of an ultrafilter all of whose members are IP sets adds no first-order consequences to ATR0\mathbf{ATR_0}; the source presents the claim in the context of the difficulty of separating Hindman's-type principles from ACA0\mathbf{ACA_0}, and gives no resolution.

Sources & referencesView supporting material

Primary source

Henry Towsner, “Ultrafilters in Reverse Mathematics”, arXiv:1109.3902 (2011).

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