The Urysohn Lemma is independent of ZF + Countable Choice
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- by Eleftherios Tachtsis PDF
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Abstract:
We prove that it is relatively consistent with $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice ($\mathsf {AC}$)) that the Axiom of Countable Choice ($\mathsf {AC}^{\aleph _{0}}$) is true, but the Urysohn Lemma ($\mathsf {UL}$), and hence the Tietze Extension Theorem ($\mathsf {TET}$), is false. This settles the corresponding open problem in P. Howard and J. E. Rubin [Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, Vol. 59, American Mathematical Society, Providence, RI, 1998].
We also prove that in Läuchli’s permutation model of $\mathsf {ZFA}$ $+$ $\neg \mathsf {UL}$, $\mathsf {AC}^{\aleph _{0}}$ is false. This fills the gap in information in the above monograph of Howard and Rubin.
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Additional Information
- Eleftherios Tachtsis
- Affiliation: Department of Statistics & Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece
- MR Author ID: 657401
- Email: ltah@aegean.gr
- Received by editor(s): July 16, 2018
- Received by editor(s) in revised form: December 30, 2018
- Published electronically: May 1, 2019
- Communicated by: Heike Mildenberger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 4029-4038
- MSC (2010): Primary 03E25; Secondary 03E35, 54D10, 54D15
- DOI: https://doi.org/10.1090/proc/14590
- MathSciNet review: 3993794