David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 56 (2):592-607 (1991)
We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof of Vopěnka and Hrbáček used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in L[ 2 ω ], there cannot exist a θ-compact cardinal less than θ (where θ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form L[ A ]. The result for L[ 2 ω ] is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Mitchell Spector (1991). Iterated Extended Ultrapowers and Supercompactness Without Choice. Annals of Pure and Applied Logic 54 (2):179-194.
Similar books and articles
G. P. Monro (1983). On Generic Extensions Without the Axiom of Choice. Journal of Symbolic Logic 48 (1):39-52.
J. M. Henle (1984). Spector Forcing. Journal of Symbolic Logic 49 (2):542-554.
Paul E. Howard (1973). Limitations on the Fraenkel-Mostowski Method of Independence Proofs. Journal of Symbolic Logic 38 (3):416-422.
J. A. Makowsky (1985). Vopěnka's Principle and Compact Logics. Journal of Symbolic Logic 50 (1):42-48.
Lorenz Halbeisen & Saharon Shelah (2001). Relations Between Some Cardinals in the Absence of the Axiom of Choice. Bulletin of Symbolic Logic 7 (2):237-261.
Bernd I. Dahn (1979). Constructions of Classical Models by Means of Kripke Models (Survey). Studia Logica 38 (4):401 - 405.
Masanao Ozawa (1995). Scott Incomplete Boolean Ultrapowers of the Real Line. Journal of Symbolic Logic 60 (1):160-171.
Robert Goldblatt (1985). On the Role of the Baire Category Theorem and Dependent Choice in the Foundations of Logic. Journal of Symbolic Logic 50 (2):412-422.
Mitchell Spector (1988). Ultrapowers Without the Axiom of Choice. Journal of Symbolic Logic 53 (4):1208-1219.
Added to index2009-01-28
Total downloads7 ( #423,260 of 1,796,218 )
Recent downloads (6 months)2 ( #349,835 of 1,796,218 )
How can I increase my downloads?