A class of strong diamond principles

  • J. D. Hamkins, “A class of strong diamond principles,” Mathematics arxiv preprint http://arxiv.org/abs/math/0211419, 2002.  
    @ARTICLE{Hamkins:LaverDiamond,
    author = {Joel David Hamkins},
    title = {A class of strong diamond principles},
    journal = {Mathematics arxiv preprint http://arxiv.org/abs/math/0211419},
    year = {2002},
    eprint = {math/0211419},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

In the context of large cardinals, the classical diamond principle $\Diamond_\kappa$ is easily strengthened in natural ways. When $\kappa$ is a measurable cardinal, for example, one might ask that a $\Diamond_\kappa$ sequence anticipate every subset of $\kappa$ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function $\ell:\kappa\to V_\kappa$ such that for any $A\in H(\kappa^+)$ there is an embedding $j:V\to M$ having critical point $\kappa$ with $j(\ell)(\kappa)=A$. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.