Forcing a countable structure to belong to the ground model

Itay Kaplan*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Suppose that P is a forcing notion, L is a language (in V), (Formula presented.) a P-name such that (Formula presented.) “(Formula presented.) is a countable L-structure”. In the product P × P, there are names (Formula presented.) such that for any generic filter G = G1 × G2 over P × P, (Formula presented.) and (Formula presented.). Zapletal asked whether or not (Formula presented.) implies that there is some M ∈ V such that (Formula presented.). We answer this question negatively and discuss related issues.

Original languageAmerican English
Pages (from-to)530-546
Number of pages17
JournalMathematical Logic Quarterly
Issue number6
StatePublished - 1 Dec 2016

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation (grant no. 1533/14). This research has received funding from the European Research Council, ERC Grant Agreement no. 338821. This is publication no. 1054 on the second author's list of publications. We thank the anonymous referee for his or her very thorough report and for his or her useful suggestions and Antonio Montalban for bringing to our attention that another paper released recently has some similar results.

Publisher Copyright:
© 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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