Power-like models of set theory

Journal of Symbolic Logic 66 (4):1766-1782 (2001)
  Copy   BIBTEX


A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| for each a ∈ M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ 1 -like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal θ such that $2^{ then every consistent extension of ZF also has a θ + -like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ 2 -like model. Here we prove: THEOREM A. If θ has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a θ-like model. (ii) $\Phi \subseteq T$ , where Φ is the recursive set of axioms {∃ κ(κ is n-Mahlo and "V κ is a Σ n -elementary submodel of the universe"): n ∈ ω}. (iii) T has a λ-like model for every uncountable cardinal λ. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an ω-Mahlo cardinal". (ii) "For every finite language L, all ℵ 2 -like models of ZFC(L) satisfy the scheme Φ(L)



    Upload a copy of this work     Papers currently archived: 92,261

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

Proper classes via the iterative conception of set.Mark F. Sharlow - 1987 - Journal of Symbolic Logic 52 (3):636-650.
Elementary extensions of countable models of set theory.John E. Hutchinson - 1976 - Journal of Symbolic Logic 41 (1):139-145.
Leibnizian models of set theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
The spectrum of resplendency.John T. Baldwin - 1990 - Journal of Symbolic Logic 55 (2):626-636.
Nonexistence of universal orders in many cardinals.Menachem Kojman & Saharon Shelah - 1992 - Journal of Symbolic Logic 57 (3):875-891.
Blunt and topless end extensions of models of set theory.Matt Kaufmann - 1983 - Journal of Symbolic Logic 48 (4):1053-1073.


Added to PP

57 (#282,512)

6 months
18 (#144,337)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Citations of this work

Rank-initial embeddings of non-standard models of set theory.Paul Kindvall Gorbow - 2020 - Archive for Mathematical Logic 59 (5-6):517-563.
Model theory of the regularity and reflection schemes.Ali Enayat & Shahram Mohsenipour - 2008 - Archive for Mathematical Logic 47 (5):447-464.
On Keisler singular‐like models.Shahram Mohsenipour - 2008 - Mathematical Logic Quarterly 54 (3):330-336.

Add more citations

References found in this work

Model Theory.Michael Makkai, C. C. Chang & H. J. Keisler - 1991 - Journal of Symbolic Logic 56 (3):1096.
The Higher Infinite.Akihiro Kanamori - 2000 - Studia Logica 65 (3):443-446.
The tree property at successors of singular cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Aronszajn trees on ℵ2 and ℵ3.Uri Abraham - 1983 - Annals of Mathematical Logic 24 (3):213-230.

View all 7 references / Add more references