The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova.

Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at $\kappa^{++}$, assuming that $\kappa=\kappa^{<\kappa}$ and there is a weakly compact cardinal above $\kappa$.
If in addition $\kappa$ is supercompact then we can force $\kappa$ to be $\aleph_\omega$ in the extension. The proofs combine the techniques of adding and then destroying a non-reflecting stationary set or a $\kappa^{++}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into $\aleph_\omega$.

Downloads:

[No arXiv entry][No published version][No entry on mathscinet][No related presentations][No related material]

This entry was posted in Compactness and tagged , , , . Bookmark the permalink.

One Response to The eightfold way

  1. saf says:

    Submitted to J. Symbolic Logic, December 2016.
    Accepted November 2017.

       0 likes

Leave a Reply

Your email address will not be published. Required fields are marked *