The bounded proper forcing axiom
The bounded proper forcing axiom
复制标题
有界真强制公理
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
S. Shelah
中科院分区：
文献类型：
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作者：
M. Goldstern;S. Shelah
有界真强制公理BPFA是这样的命题：对于任意一个真强制概念的极大反链族，每个极大反链的大小都是1，都有一个有向集满足所有这些反链。正则基数κ称为∑1-反射，如果对任意正则基数χ，对所有公式φ，“H（χ）<$'<$'”蕴涵“<$δ < κ，H（δ）<$'<$'”.我们研究了BPFA的几个代数结果，证明了有界真强迫公理的相容性强度恰好是∑1-反射基数的存在性（小于Mahlo基数的存在性）.我们还表明，两个结构之间的同构存在的问题可以减少到一个结构的刚性问题。
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.