We show that a strong form of the so called Lindström’s Theorem [4] fails to generalize to extensions of L κ ω and L κ κ : For weakly compact κ there is no strongest extension of L κ ω with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to κ. With an additional set-theoretic assumption, there is no strongest extension of L κ κ with the (κ,κ)-compactness property and the Löwenheim-Skolem theorem down to <κ.

We show that form, n≧1 the existence of a∏ n m indescribable cardinal is equiconsistent with the failure of the combinatorial principle at a∏ n m indescribable cardinal κ together with the Generalized Continuum Hypothesis.

Unfoldable cardinals are preserved by fast function forcing and the Laver-like preparations that fast functions support. These iterations show, by set-forcing over any model of ZFC, that any given unfoldable cardinal κ can be made indestructible by the forcing to add any number of Cohen subsets to κ.

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to (...) add a Cohen subset to κ, the forcing to shoot a club Cκ avoiding the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context. (shrink)