Hostname: page-component-cc8bf7c57-7lvjp Total loading time: 0 Render date: 2024-12-10T15:55:54.864Z Has data issue: false hasContentIssue false

On a generalization of Jensen's □κ, and strategic closure of partial orders

Published online by Cambridge University Press:  12 March 2014

Dan Velleman*
Affiliation:
University of Texas, Austin, Texas 78712

Extract

It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □κ. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □κ.

In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈P, ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈pββ < α〉 Good chooses all terms pβ for limit β, and Bad chooses all the others. Bad wins if for some limit β<α, Good is unable to move at stage β because 〈pγγ < β〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beller, A. and Litman, A., A strengthening of Jensen's β principles, this Journal, vol. 45 (1980), pp. 251264.Google Scholar
[2]Jensen, R.B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.CrossRefGoogle Scholar
[3]Prikry, K. and Solovay, R.M., On partitions into stationary sets, this Journal, vol. 40 (1975), pp. 7580.Google Scholar
[4]Shelah, S. and Stanley, L., S-forcing, I: A “black box” theorem for morasses, with applications to super-Souslin trees, Israel Journal of Mathematics, vol. 43 (1982), pp. 185224.CrossRefGoogle Scholar
[5]Velleman, D., Morasses, diamond, and forcing, Annals of Mathematical Logic (to appear).Google Scholar