Notre Dame Journal of Formal Logic

Powers of 2

Horst Herrlich and Kyriakos Keremedis


It is shown that in ZF Martin's $ \aleph_{0}^{}$-axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2X of a 2-point discrete space are Baire; and that the latter property implies the following: (a) the axiom of countable choice for finite sets, (b) power sets of infinite sets are Dedekind-infinite, (c) there are no amorphous sets, and (d) weak forms of the Kinna-Wagner principle.

Article information

Notre Dame J. Formal Logic, Volume 40, Number 3 (1999), 346-351.

First available in Project Euclid: 28 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E25: Axiom of choice and related propositions
Secondary: 54E52: Baire category, Baire spaces


Keremedis, Kyriakos; Herrlich, Horst. Powers of 2. Notre Dame J. Formal Logic 40 (1999), no. 3, 346--351. doi:10.1305/ndjfl/1022615615.

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  • Blass, A., “Injectivity, projectivity, and the axiom of choice,” Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31–59. Zbl 0426.03053 MR 81j:04004
  • Brunner, N., “Sequential compactness and the axiom of choice,” Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 89–92. Zbl 0502.03030 MR 84f:04009
  • Fossy, J., and M. Morillon, “The Baire category property and some notions of compactness,” Journal of the London Mathematical Society, vol. 57 (1998), pp. 1–19. Zbl 0922.03070 MR 99m:03096
  • Herrlich, H., and K. Keremedis, “Products, the Baire category theorem, and the axiom of dependent choice,” Commentationes Mathematicae Universitatis Carolinae, vol. 40 (1999), pp. 771–75. MR 2001e:03089
  • Howard P., and J. E. Rubin, “Consequences of the Axiom of Choice,” American Mathematical Society, Mathematical Surveys and Monographs, vol. 59 (1998). Zbl 0947.03001 MR 99h:03026
  • Łos, J., and C. Ryll-Nardzewski, “Effectiveness of the representation theory for Boolean algebras,” Fundamenta Mathematicæ, vol. 42 (1955), pp. 49–56. MR 16:439g
  • Mycielski, J., “Two remarks on Tychonoff's product theorem,” Bulletin L'Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 12 (1964), pp. 439–41. Zbl 0138.17703 MR 35:6566
  • Rubin, H. and D. Scott, “Some topological theorems equivalent to the Boolean prime ideal theorem,” Bulletin of the American Mathematical Society, vol. 60 (1954), p. 389.
  • Shannon, G. P., “Provable forms of Martin's axiom,” Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 382–88. Zbl 0731.03026 MR 91g:03107