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The bounded proper forcing axiom

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


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.



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[To]Todorcevic, Stevo, A note on the proper forcing axiom, Axiomatic set theory (Baumgartner, 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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