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Measure theory and higher order arithmetic

Author: Alexander P. Kreuzer
Journal: Proc. Amer. Math. Soc. 143 (2015), 5411-5425
MSC (2010): Primary 03F35, 03B30; Secondary 03E35
Published electronically: April 14, 2015
MathSciNet review: 3411156
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Abstract: We investigate the statement that the Lebesgue measure defined on all subsets of the Cantor-space exists. As a base system we take $ \textsf {ACA}_0^\omega +(\mu )$. The system $ \textsf {ACA}_0^\omega $ is the higher order extension of Friedman's system $ \mathsf {ACA_0}$, and $ (\mu )$ denotes Feferman's $ \mu $, that is, a uniform functional for arithmetical comprehension defined by $ f(\mu (f))=0$ if $ \Exists {n} f(n)=0$ for $ f\in \mathbb{N}^{\mathbb{N}}$. Feferman's $ \mu $ will provide countable unions and intersections of sets of reals and is, in fact, equivalent to this. For this reason $ \mathsf {ACA_0}^\omega + (\mu )$ is the weakest fragment of higher order arithmetic where $ \sigma $-additive measures are directly definable.

We obtain that over $ \mathsf {ACA_0}^\omega + (\mu )$ the existence of the Lebesgue measure is $ \Pi ^1_2$-conservative over $ \mathsf {ACA_0}^\omega $ and with this conservative over $ \mathsf {PA}$. Moreover, we establish a corresponding program extraction result.

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Additional Information

Alexander P. Kreuzer
Affiliation: Department of Mathematics, Faculty of Science, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore 119076

Received by editor(s): June 7, 2014
Received by editor(s) in revised form: October 27, 2014
Published electronically: April 14, 2015
Additional Notes: The author was partly supported by the RECRE project, and the Ministry of Education of Singapore through grant R146-000-184-112 (MOE2013-T2-1-062).
Communicated by: Mirna Dzamonja
Article copyright: © Copyright 2015 American Mathematical Society

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