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The bounded proper forcing axiom

  • Martin Goldstern (a1) and Saharon Shelah (a2)

Abstract

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.

A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.

We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).

We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

Copyright

References

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[Ba 1]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.
[Ba 2]Baumgartner, James E., Iterated forcing, Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 159.
[Fu]Fuchino, Sakaé, On potential embedding and versions of Martin's axiom, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 481492.
[Mi]Mitchell, William, Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (19721973), pp. 2146.
[Sh 56]Shelah, Saharon, Refuting Ehrenfeucht conjecture on rigid models, Israel Journal of Mathematics, vol. 25 (1976); [= Abraham Robinson Memorial Symposium, Yale, 1975], pp. 273–286.
[Sh 73]Shelah, Saharon, Models with second-order properties, II: Trees with no undefined branches, Annals of Mathematical Logic, vol. 14 (1978), pp. 7387.
[Sh b]Shelah, Saharon, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.
[Sh f]Shelah, Saharon, Proper and improper forcing, Perspectives in Mathematical Logic, Springer Verlag.
[To]Todorcevic, Stevo, A note on the proper forcing axiom, Axiomatic set theory (Baumgartner, J.et al., editors), American Mathematical Society, Providence, Rhode Island, 1984, pp. 209218.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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