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
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by Samuel M. Corson PDF

Proc. Amer. Math. Soc. 148 (2020), 5381-5386 Request permission

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.

References

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, DOI 10.1090/conm/558/11060
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, DOI 10.1090/S0002-9939-98-04163-X
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
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, DOI 10.1002/(SICI)1521-3870(200005)46:2<219::AID-MALQ219>3.0.CO;2-2
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, DOI 10.1090/surv/059
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, DOI 10.1007/s00039-005-0503-1
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, DOI 10.1002/1521-3870(200101)47:1<117::AID-MALQ117>3.0.CO;2-N
P. Kleppmann, Free groups and the axiom of choice, PhD thesis, University of Cambridge, 2015.
Jaroslav Ne et il, Metric spaces are Ramsey, European J. Combin. 28 (2007), no. 1, 457–468. MR 2261831, DOI 10.1016/j.ejc.2004.11.003
David Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, J. Symbolic Logic 37 (1972), 721–743. MR 337605, DOI 10.2307/2272420
David Pincus, Adding dependent choice, Ann. Math. Logic 11 (1977), no. 1, 105–145. MR 453529, DOI 10.1016/0003-4843(77)90011-0
David Pincus, Adding dependent choice to the prime ideal theorem, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. Math., Vol. 87, North-Holland, Amsterdam, 1977, pp. 547–565. MR 0480027
A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977–982. MR 26802, DOI 10.1090/S0002-9904-1948-09118-2
Eleftherios Tachtsis, Disasters in metric topology without choice, Comment. Math. Univ. Carolin. 43 (2002), no. 1, 165–174. MR 1903316
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, DOI 10.1090/S0065-9266-10-00586-7