Dowker and super-Dowker filters
HTML articles powered by AMS MathViewer
- by James Cummings and Charles Morgan PDF
- Proc. Amer. Math. Soc. 145 (2017), 5381-5390 Request permission
Abstract:
Our main results show that a very simple forcing construction can be used to add Dowker and super-Dowker filters:
Let $\kappa$ be uncountable with $\kappa ^{ < \kappa }=\kappa$. Let $G$ be generic over $V$ for $Add(\kappa , \kappa ^{++})$. Then in $V[G]$ there is a Dowker filter on $\kappa ^+$.
Let $V$ be Laver’s model in which $\kappa$ is supercompact and the supercompactness of $\kappa$ is indestructible under $\kappa$-directed closed forcing, and let $G$ be generic for $Add(\kappa , \kappa ^{++})$. Then in $V[G]$ there is a super-Dowker filter on $\kappa ^+$.
References
- Z. Balogh and G. Gruenhage, On a problem of C. H. Dowker, J. Symbolic Logic 56 (1991), no. 4, 1284–1289. MR 1136457, DOI 10.2307/2275475
- James Cummings, Iterated forcing and elementary embeddings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 775–883. MR 2768691, DOI 10.1007/978-1-4020-5764-9_{1}3
- C. H. Dowker, A problem in set theory, J. London Math. Soc. 27 (1952), 371–374. MR 47741, DOI 10.1112/jlms/s1-27.3.371
- Chris Freiling and T. H. Payne, Some properties of large filters, J. Symbolic Logic 53 (1988), no. 4, 1027–1035. MR 973098, DOI 10.2307/2274602
- Richard Laver, Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing, Israel J. Math. 29 (1978), no. 4, 385–388. MR 472529, DOI 10.1007/BF02761175
Additional Information
- James Cummings
- Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 289375
- ORCID: 0000-0002-7913-0427
- Email: jcumming@andrew.cmu.edu
- Charles Morgan
- Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
- MR Author ID: 290043
- Email: charles.morgan@ucl.ac.uk
- Received by editor(s): June 20, 2016
- Received by editor(s) in revised form: January 12, 2017
- Published electronically: August 7, 2017
- Additional Notes: The first author was partially supported by NSF grant DMS-1500790.
- Communicated by: Mirna Džamonja
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5381-5390
- MSC (2010): Primary 03E35; Secondary 03E55, 03E05
- DOI: https://doi.org/10.1090/proc/13706
- MathSciNet review: 3717964