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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Measure theory and higher order arithmetic
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by Alexander P. Kreuzer PDF
Proc. Amer. Math. Soc. 143 (2015), 5411-5425 Request permission

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.

References
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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
  • Email: matkaps@nus.edu.sg
  • 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 Dz̆amonja
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 5411-5425
  • MSC (2010): Primary 03F35, 03B30; Secondary 03E35
  • DOI: https://doi.org/10.1090/proc/12671
  • MathSciNet review: 3411156