ON BOX, WEAK BOX AND STRONG COMPACTNESS. ARTHUR W. APTER AND JAMES M. HENLE. One of the most important goals of set theorists over the ...
ON BOX, WEAK BOX AND STRONG COMPACTNESS ARTHUR W. APTER AND JAMES M. HENLE One of the most important goals of set theorists over the last few years has been to re-prove old results which previously had used very strong assumptions from hypotheses which, at least prima facie, are weaker. Examples of these abound, including, but certainly not limited to, the work of Woodin and Cummings (see [3]) on the Singular Cardinals Problem, in which results previously obtained by Magidor[5, 6] using supercompactness and hugeness were re-proven using hyper-
measurability. This paper continues the work of [1] along these lines, re-proving a result of Ben-David and Magidor[2] using strong compactness instead of supercompactness. In [2], the authors show that Con (ZFC + GCH + 3K[K is K+ supercompact]) => Con (ZFC +•«„, + "' DKJ, where • * and • * are, respectively, the combinatorial principles box and weak box at the cardinal K. (See [2] for the appropriate definitions.) We prove the following. THEOREM.
Con (ZFC + GCH + 3K[K is
K+
strongly compact)] => Con (ZFC +
The proof recasts the ideas of [2] and [5] in terms of strong compactness. As such, we are going to assume complete familiarity with the ideas and techniques of [2] and [5]. Our notation and terminology will generally be that of [1], [2] and [5]. V will be our ground model. We recall that a cardinal K is K+ strongly compact if and only if PK(K+) carries a «r-additive fine ultrafilter U. Although U cannot be assumed to be normal, as mentioned in [1], U can be chosen so as to have some weak normality properties. Specifically, let k:PK(ic+) -> K be defined by k(p) = p n K. Let \JK = k+(\J) be the Rudin-Keisler projection to an ultrafilter on K, that is, xe\JK if and only if k~1(x)e\J. In [1], it is shown that U can be chosen such that \JK is a normal measure on K, and such that U is weakly normal in the sense that if/: PK{K+) -> K is a choice function (f(p)ep) then/is constant on a set in U. This means that D = {pePK(K+): p n K is an inaccessible cardinal} e U. We are now in a position to define the partial ordering P used in the proof of our Theorem. The definition is an amalgamation of the ideas of [1] with the corresponding definition of [2] and [5]. Specifically, P is the set of all finite tuples of the form satisfying the following properties. (p1,...,pn,f0,...,fn,A,F) 1. For 1 ^ / < «, Pi^D and pt 0 K )» Pn f) K < q 0 K, then/„eCol(K+,q n K). 5. Let Dp = {qsD:pn n K < q (1 K). Fis a function with domain [Dp ]/,»,^?-O be a condition of length n withj ^ n, and let
