Dense ideals and Cardinal Arithmetic

From large cardinals we show the consistency of normal, fine, κ-complete -dense ideals on Pκ( ) for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman. Most large cardinals are characterizable in terms of elementary embeddings between models of set theory that have a certain amount of agreement with the full universeV . A typical large cardinal is the least ordinalmoved by a nontrivialmap j : V → M , whereM is a transitive class, and the strengthof the large cardinal assumption tends to increase asM gets closer toV . Such cardinals are inaccessible andmuch more. This phenomenon can however be realized at small cardinals when the embedding j : V → M is defined in a forcing extension V [G ]. The nature of the forcing adds another dimension to these “generic large cardinals,” and their strength tends to increase as the three models V ,M , andV [G ] more closely resemble one another. Here, we consider generic versions of supercompactness at successor cardinals that are optimal in the sense that the forcing poset needed to produce the elementary embedding is of the smallest possible size. We show that relative to a super-almosthuge cardinal, there can exist a successor cardinal κ such that for every regular ≥ κ, there is a normal, fine, κ-complete, -dense ideal on Pκ( ). As far as the author knows, this is the first result exhibiting the consistency of even saturated normal and fine ideals on Pκ( ) for a fixed successor κ and several values of simultaneously. The method used also has immediate application to show the nonabsoluteness of some cardinal characteristics of the powerset of a fixed regular cardinal , even between models with the same cardinals and same -sequences. Generic large cardinals can have strong influence over the combinatorial structure of the universe in their vicinity. We explore the interplay between dense ideals, cardinal arithmetic, nonregular ultrafilters, and stationary reflection. We answer two open questions posed by Foreman in [7] and provide a “global” counterexample to an old conjecture in model theory. We also show some limitations of dense ideals near singular cardinals, establishing the optimality some aspects of our consistency results. Finally we show that in contrast to traditional supercompactness, the strong forms of generic supercompactness considered here are compatible with Jensen’s square principle. Received December 6, 2014. 2010Mathematics Subject Classification. Primary 03E35, Secondary 03E55.