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On the equivalence of certain consequences of the proper forcing axiom

  • Peter Nyikos (a1) and Leszek Piątkiewicz (a2)

Abstract

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω1 with ω1 generators, then there exists an uncountable Xω1, such that either [X]ωI = ∅ or [X]ωI.

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[B] Baumgartner, James E., Applications of the Proper Forcing Axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 913959.
[vD] van Douwen, Eric K., The integers and topology, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 111167.
[F] Fremlin, D. H., Consequences of Martin's axiom, Cambridge University Press, Cambridge, 1984.
[N] Nyikos, Peter, Applications of the Proper Forcing Axiom, preprint.
[R] Roitman, Judy, Basic S and L, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland,, Amsterdam, 1984, pp. 295326.
[T] Todorĉević, Stevo, Forcing positive partition relations, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 703720.
[V] Vaughan, Jerry E., Small uncountable cardinals and topology. With an appendix by S. Shelah, Open problems in topology (van Mill, J. and Reed, G. M., editors), North-Holland, Amsterdam, 1990, pp. 195218.
[W] Weiss, William, Versions of Martin's axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 827886.

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On the equivalence of certain consequences of the proper forcing axiom

  • Peter Nyikos (a1) and Leszek Piątkiewicz (a2)

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