Open Access
2011 On Absoluteness of Categoricity in Abstract Elementary Classes
Sy-David Friedman, Martin Koerwien
Notre Dame J. Formal Logic 52(4): 395-402 (2011). DOI: 10.1215/00294527-1499354


Shelah has shown that $\aleph_1$-categoricity for Abstract Elementary Classes (AECs) is not absolute in the following sense: There is an example $K$ of an AEC (which is actually axiomatizable in the logic $L(Q)$) such that if $2^{\aleph_0}<2^{\aleph_1}$ (the weak CH holds) then $K$ has the maximum possible number of models of size $\aleph_1$, whereas if Martin's Axiom at $\aleph_1$ (denoted by MA) holds then $K$ is $\aleph_1$-categorical. In this note we extract the properties from Shelah's example which make both parts work resulting in our definitions of condition A and condition B, and then we show that for any AEC satisfying these two conditions, neither of these implications can be reversed.


Download Citation

Sy-David Friedman. Martin Koerwien. "On Absoluteness of Categoricity in Abstract Elementary Classes." Notre Dame J. Formal Logic 52 (4) 395 - 402, 2011.


Published: 2011
First available in Project Euclid: 4 November 2011

zbMATH: 1252.03093
MathSciNet: MR2855878
Digital Object Identifier: 10.1215/00294527-1499354

Primary: 03C35 , 03C52 , 03E35

Keywords: absoluteness , abstract elementary classes , categoricity , Forcing

Rights: Copyright © 2011 University of Notre Dame

Vol.52 • No. 4 • 2011
Back to Top