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Simple forcing notions and forcing axioms

  • Andrzej Rosłanowski (a1) (a2) and Saharon Shelah (a1) (a3)

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In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [4] in which several problems were posed. We answer some of those problems here.

In the first section we deal with the problem of adding Cohen reals by simple forcing notions. Here we interpret simple as of small size. We try to establish as weak as possible versions of Martin Axiom sufficient to conclude that some forcing notions of size less than the continuum add a Cohen real. For example we show that MA(σ-centered) is enough to cause that every small σ-linked forcing notion adds a Cohen real (see Theorem 1.2) and MA(Cohen) implies that every small forcing notion adding an unbounded real adds a Cohen real (see Theorem 1.6). A new almost ωω-bounding σ-centered forcing notion ℚ appears naturally here. This forcing notion is responsible for adding unbounded reals in this sense, that MA(ℚ) implies that every small forcing notion adding a new real adds an unbounded real (see Theorem 1.13).

In the second section we are interested in Anti-Martin Axioms for simple forcing notions. Here we interpret simple as nicely definable. Our aim is to show the consistency of AMA for as large as possible class of ccc forcing notions with large continuum. It has been known that AMA(ccc) implies CH, but it has been (rightly) expected that restrictions to regular (simple) forcing notions might help.

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[1]Bartoszyński, Tomek and Judah, Haim, Set theory: On the structure of the real line, A. K. Peters, Wellesley, Massachusetts, 1995.
[2]Cichoń, Jacek, Anti-Martin axiom, circulated notes, 1989.
[3]Jech, Thomas, Set theory, Academic Press, 1978.
[4]Judah, Haim and Rosłanowski, Andrzej, Martin axiom and the continuum, this Journal, vol. 60 (1995), pp. 374391.
[5]Miller, Arnold and Prikry, Karel, When the continuum has cofinality ω 1, Pacific Journal of Mathematics, vol. 115 (1984), pp. 399407.
[6]Shelah, Saharon, Proper and improper forcing, Perspectives in mathematical logic, Springer-Verlag, accepted.
[7]Shelah, Saharon, How special are Cohen and random forcings, Israel Journal of Mathematics, vol. 88 (1994), pp. 153174.
[8]Todorcevic, Stevo, Remarks on Martin's axiom and the continuum hypothesis, Canadian Journal of Mathematics, vol. 43 (1991), pp. 832851.
[9]van Douwen, Eric K. and Fleissner, William G., Definable forcing axiom: An alternative to Martin's axiom, Topology and its Applications, vol. 35 (1990), pp. 277289.

Simple forcing notions and forcing axioms

  • Andrzej Rosłanowski (a1) (a2) and Saharon Shelah (a1) (a3)

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