The independence of Stone’s Theorem from the Boolean Prime Ideal Theorem

Corson, Samuel

doi:10.1090/proc/15164

The independence of Stone's Theorem from the Boolean Prime Ideal Theorem

Author: Samuel M. Corson
Journal: Proc. Amer. Math. Soc. 148 (2020), 5381-5386
MSC (2010): Primary 03E25, 54A35, 54E35, 54D20
DOI: https://doi.org/10.1090/proc/15164
Published electronically: September 18, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a permutation model in which Stone's theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree, and Watson.

[1] Andreas Blass, Partitions and permutation groups, Model theoretic methods in finite combinatorics, Contemp. Math., vol. 558, Amer. Math. Soc., Providence, RI, 2011, pp. 453–466. MR 3418642, https://doi.org/10.1090/conm/558/11060
[2] C. Good, I. J. Tree, and W. S. Watson, On Stone’s theorem and the axiom of choice, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1211–1218. MR 1425122, https://doi.org/10.1090/S0002-9939-98-04163-X
[3] J. D. Halpern and A. Lévy, The Boolean prime ideal theorem does not imply the axiom of choice., Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 83–134. MR 0284328
[4] Paul Howard, Kyriakos Keremedis, Jean E. Rubin, and Adrienne Stanley, Paracompactness of metric spaces and the axiom of multiple choice, MLQ Math. Log. Q. 46 (2000), no. 2, 219–232. MR 1755811, https://doi.org/10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2
[5] Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998. With 1 IBM-PC floppy disk (3.5 inch; WD). MR 1637107
[6] A. S. Kechris, V. G. Pestov, and S. Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), no. 1, 106–189. MR 2140630, https://doi.org/10.1007/s00039-005-0503-1
[7] Kyriakos Keremedis and Eleftherios Tachtsis, Compact metric spaces and weak forms of the axiom of choice, MLQ Math. Log. Q. 47 (2001), no. 1, 117–128. MR 1808950, https://doi.org/10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N
[8] P. Kleppmann, Free groups and the axiom of choice, PhD thesis, University of Cambridge, 2015.
[9] Jaroslav Nešetřil, Metric spaces are Ramsey, European J. Combin. 28 (2007), no. 1, 457–468. MR 2261831, https://doi.org/10.1016/j.ejc.2004.11.003
[10] David Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, J. Symbolic Logic 37 (1972), 721–743. MR 337605, https://doi.org/10.2307/2272420
[11] David Pincus, Adding dependent choice, Ann. Math. Logic 11 (1977), no. 1, 105–145. MR 453529, https://doi.org/10.1016/0003-4843(77)90011-0
[12] David Pincus, Adding dependent choice to the prime ideal theorem, Logic Colloquium 76 (Oxford, 1976) North-Holland, Amsterdam, 1977, pp. 547–565. Studies in Logic and Found. Math., Vol. 87. MR 0480027
[13] A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977–982. MR 26802, https://doi.org/10.1090/S0002-9904-1948-09118-2
[14] Eleftherios Tachtsis, Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 165–174. MR 1903316
[15] L. Nguyen Van Thé, Structural Ramsey theory of metric spaces and topological dynamics of isometry groups, Mem. Amer. Math. Soc. 206 (2010), no. 968, x+140. MR 2667917, https://doi.org/10.1090/S0065-9266-10-00586-7