Sets constructible from sequences of ultrafilters

Journal of Symbolic Logic 39 (1):57-66 (1974)
  Copy   BIBTEX

Abstract

In [4], Kunen used iterated ultrapowers to show that ifUis a normalκ-complete nontrivial ultrafilter on a cardinalκthenL[U], the class of sets constructive fromU, has only the ultrafilterU∩L[U] and this ultrafilter depends only onκ. In this paper we extend Kunen's methods to arbitrary sequencesUof ultrafilters and obtain generalizations of these results. In particular we answer Problem 1 of Kunen and Paris [5] which asks whether the number of ultrafilters onκcan be intermediate between 1 and 22κ. If there is a normalκ-complete ultrafilterUonκsuch that {α <κ: α is measurable} ∈Uthen there is an inner model with exactly two normal ultrafilters onκ, and ifκis super-compact then there are inner models havingκ+ +,κ+or any cardinal less than or equal toκnormal ultrafilters.These methods also show that several properties ofLwhich had been shown to hold forL[U] also hold forL[U]: using an idea of Silver we show that inL[U] the generalized continuum hypothesis is true, there is a Souslin tree, and there is awell-ordering of the reals. In addition we generalize a result of Kunen to characterize the countaby complete ultrafilters ofL[U].

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 76,199

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Ultrafilters which extend measures.Michael Benedikt - 1998 - Journal of Symbolic Logic 63 (2):638-662.
Saturating ultrafilters on N.D. H. Fremlin & P. J. Nyikos - 1989 - Journal of Symbolic Logic 54 (3):708-718.
A characterization of 2-square ultrafilters.Ned I. Rosen - 1983 - Journal of Symbolic Logic 48 (2):409-414.
Ultrafilters on the natural numbers.Christopher Barney - 2003 - Journal of Symbolic Logic 68 (3):764-784.
Sets constructed from sequences of measures: Revisited.William J. Mitchell - 1983 - Journal of Symbolic Logic 48 (3):600-609.
On Milliken-Taylor Ultrafilters.Heike Mildenberger - 2011 - Notre Dame Journal of Formal Logic 52 (4):381-394.
Reasonable Ultrafilters, Again.Andrzej Rosłanowski & Saharon Shelah - 2011 - Notre Dame Journal of Formal Logic 52 (2):113-147.
Ultrafilters on ω.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.

Analytics

Added to PP
2009-01-28

Downloads
41 (#286,677)

6 months
1 (#448,894)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

Strong axioms of infinity and elementary embeddings.Robert M. Solovay - 1978 - Annals of Mathematical Logic 13 (1):73.
Descriptive inner model theory.Grigor Sargsyan - 2013 - Bulletin of Symbolic Logic 19 (1):1-55.
The consistency strength of projective absoluteness.Kai Hauser - 1995 - Annals of Pure and Applied Logic 74 (3):245-295.
Ultrafilters over a measurable cardinal.A. Kanamori - 1976 - Annals of Mathematical Logic 10 (3-4):315-356.
Sets in Prikry and Magidor generic extensions.Tom Benhamou & Moti Gitik - 2021 - Annals of Pure and Applied Logic 172 (4):102926.

View all 26 citations / Add more citations

References found in this work

The fine structure of the constructible hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
Some applications of iterated ultrapowers in set theory.Kenneth Kunen - 1970 - Annals of Mathematical Logic 1 (2):179.
Boolean extensions and measurable cardinals.K. Kunen - 1971 - Annals of Mathematical Logic 2 (4):359.

Add more references