Measure theory and higher order arithmetic [article]

Alexander P. Kreuzer
2015 arXiv   pre-print
We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor space exists. As base system we take ACA_0^ω + (μ). The system ACA_0^ω is the higher order extension of Friedman's system ACA_0, and (μ) denotes Feferman's μ, that is a uniform functional for arithmetical comprehension defined by f(μ(f))=0 if ∃ n f(n)=0 for f∈N^N. Feferman's μ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reasons ACA_0^ω +
more » ... μ) is the weakest fragment of higher order arithmetic where σ-additive measures are directly definable. We obtain that over ACA_0^ω + (μ) the existence of the Lebesgue measure is Π^1_2-conservative over ACA_0^ω and with this conservative over PA. Moreover, we establish a corresponding program extraction result.
arXiv:1312.1531v2 fatcat:tctxumkoyvb6rapk3hdhc7553e