Proper forcing axiom and selective separability
نویسندگان
چکیده
منابع مشابه
The Proper Forcing Axiom
The Proper Forcing Axiom is a powerful extension of the Baire Category Theorem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the Continuum Hypothesis, it eliminates a large number of the pathological constructions which can be carried out using additional axioms of set theory. Mathematics Subject Classification (2000). Primary...
متن کاملThe Bounded Proper Forcing Axiom
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 Σ1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) |= ‘φ’ ” implies “∃δ<κ, H(δ) |= ‘φ’ ” We show that BPFA is equivalent to the statement tha...
متن کاملThe Proper Forcing Axiom: a tutorial
In these notes we will present an exposition of the Proper Forcing Axiom (PFA). We will first discuss examples of the consequences of PFA. We will then present two proper partial orders which are used to force two combinatorial principles which follow from PFA: The P-Ideal Dichotomy (PID) and Todorcevic’s formulation of the Open Coloring Axiom. On one hand, these posets and the proofs of their ...
متن کاملBounded Proper Forcing Axiom and Well Orderings of the Reals
We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω1) which is ∆1 definable with parameter a subset of ω1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes א2 and also satisfies BPFA must contain all subsets of ω1. We show as applications how to build minimal models of BPFA and that BPFA implies t...
متن کاملThe proper forcing axiom and the singular cardinal hypothesis
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses ideas of Moore from [11] and the notion of a relativized trace function on pairs of ordinals.
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ژورنال
عنوان ژورنال: Topology and its Applications
سال: 2012
ISSN: 0166-8641
DOI: 10.1016/j.topol.2011.11.048