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- W2124205440 abstract "It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □ κ . The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □ κ . In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈 P , ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈 p β ∣ β < α 〉 Good chooses all terms p β for limit β , and Bad chooses all the others. Bad wins if for some limit β < α , Good is unable to move at stage β because 〈 p γ ∣ γ < β 〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good." @default.
- W2124205440 created "2016-06-24" @default.
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- W2124205440 date "1983-12-01" @default.
- W2124205440 modified "2023-10-16" @default.
- W2124205440 title "On a generalization of Jensen's □κ, and strategic closure of partial orders" @default.
- W2124205440 cites W2003575709 @default.
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- W2124205440 doi "https://doi.org/10.2307/2273668" @default.
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