Potential isomorphism and semi-proper trees

Alex Hellsten¹ ; Tapani Hyttinen¹ ; Saharon Shelah²

¹ Department of Mathematics University of Helsinki 00014 Helsinki, Finland
² Institute of Mathematics The Hebrew University 91904 Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.

Fundamenta Mathematicae, Tome 175 (2002) no. 2, pp. 127-142

Résumé

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We show that the existence of weakly semi-proper trees is consistent relative to ${\rm ZFC}$ by proving the existence of weakly semi-proper trees under certain cardinal arithmetic assumptions. We also prove the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.

Détail

DOI : 10.4064/fm175-2-3

Keywords: study notion potential isomorphism where structures said potentially isomorphic isomorphic generic extension preserves stationary sets does sets cardinality cardinality models introduce notion weakly semi proper trees note there strong connection between existence potentially isomorphic models given complete theory existence weakly semi proper trees existence weakly semi proper trees consistent relative zfc proving existence weakly semi proper trees under certain cardinal arithmetic assumptions prove consistency non existence weakly semi proper trees assuming consistency large cardinals