Uncountable Admissibles I: Forcing
نویسنده
چکیده
Assume V = L. Let k be a regular cardinal and for X Qk let a(X) denote the least ordinal a such that ¿„[A"] is admissible. In this paper we characterize those ordinals of the form k s.t. La( X) is admissible. Question (V — L). Which admissible ordinals are of the form a(X) for some X QkI We deal in this paper with the case where k is regular. (The singular case will be treated in Friedman [1981a].) The answer to this Question is best phrased in terms of a strong form of < «-closure which we call < K-admissibility. Received by the editors June 20, 1980. The results were presented at the Greek Logic Symposium, University of Patras, 1980. 1980 Mathematics Subject Classification. Primary 03D60, 03E45; Secondary 03E40. 1 The preparation of this paper was supported by a National Science Foundation grant. © 1982 American Mathematical Society 0002-9947/81 /0000-1023/$04.25 61 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
منابع مشابه
Proper forcing, cardinal arithmetic, and uncountable linear orders
In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ1-definable in (H(ω2),∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, ω∗ 1 , C, C ∗ where X is any suborder of the reals of size ...
متن کاملLocally Compact, Locally Countable Spaces and Random Reals
In this note I will present a proof that, assuming PFA, if R is a measure algebra then after forcing with R every uncountable locally compact locally countable cometrizable space contains an uncountable discrete set. The lemmas and techniques will be presented in a general form as they may be applicable to other problems.
متن کاملOn the Equivalence of Certain Consequences of the Proper Forcing Axiom
We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on co, with co, generators, then there exists an uncountable X C co,, such that either [X]w n I = 0 or ...
متن کاملThe Utility of the Uncountable
In my lecture at the 2011 Congress on Logic, Methodology, and the Philosophy of Science in Nancy, France, I spoke on an additional axiom of set theory — the Proper Forcing Axiom — which has proved very successful in settling combinatorial problems concerning uncountable sets. Since I have already written a exposition on this subject [43], I have decided to address a broader question in this art...
متن کاملUncountable superperfect forcing and minimality
Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ = κ , using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter (or p-filter) F...
متن کامل