David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 48 (4):1053-1073 (1983)
Let U be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that U has a Σ n end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If U is a well-founded model of ZFC, such that U has uncountable cofinality and U has a topless Σ 3 end extension, then U has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in U) highly hyperinaccessible. In § 3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σ n end extension for each n ∈ ω. The case κ = ω 1 is investigated more deeply in § 4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω 1 -like models of ZFC are constructed which have a Σ n end extension for all n ∈ ω but have no elementary end extension. ω 1 -like models of ZFC which have no Σ 3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions
|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
No citations found.
Similar books and articles
Ignacio Jané & Gabriel Uzquiano (2004). Well- and Non-Well-Founded Fregean Extensions. Journal of Philosophical Logic 33 (5):437-465.
Saharon Shelah & Simon Thomas (1997). The Cofinality Spectrum of the Infinite Symmetric Group. Journal of Symbolic Logic 62 (3):902-916.
Jacob Lurie (1999). Anti-Admissible Sets. Journal of Symbolic Logic 64 (2):407-435.
Andreas Blass (1974). On Certain Types and Models for Arithmetic. Journal of Symbolic Logic 39 (1):151-162.
Andrés Villaveces (1998). Chains of End Elementary Extensions of Models of Set Theory. Journal of Symbolic Logic 63 (3):1116-1136.
George Mills (1978). A Model of Peano Arithmetic with No Elementary End Extension. Journal of Symbolic Logic 43 (3):563-567.
Saharon Shelah (1978). End Extensions and Numbers of Countable Models. Journal of Symbolic Logic 43 (3):550-562.
John E. Hutchinson (1976). Elementary Extensions of Countable Models of Set Theory. Journal of Symbolic Logic 41 (1):139-145.
Andrés Villaveces (1999). Heights of Models of ZFC and the Existence of End Elementary Extensions II. Journal of Symbolic Logic 64 (3):1111-1124.
Ali Enayat (2001). Power-Like Models of Set Theory. Journal of Symbolic Logic 66 (4):1766-1782.
Sorry, there are not enough data points to plot this chart.
Added to index2009-01-28
Total downloads1 ( #402,963 of 1,096,504 )
Recent downloads (6 months)1 ( #238,630 of 1,096,504 )
How can I increase my downloads?