Forcing a countable structure to belong to the ground model

Itay Kaplan; Saharon Shelah

doi:10.1002/malq.201400094

Forcing a countable structure to belong to the ground model

Itay Kaplan^*, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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 language	American English
Pages (from-to)	530-546
Number of pages	17
Journal	Mathematical Logic Quarterly
Volume	62
Issue number	6
DOIs	https://doi.org/10.1002/malq.201400094
State	Published - 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

Access to Document

10.1002/malq.201400094

Cite this

@article{890e98eb4e5a4602a5a5f298525d9ed6,

title = "Forcing a countable structure to belong to the ground model",

abstract = "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.",

author = "Itay Kaplan and Saharon Shelah",

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: {\textcopyright} 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim",

year = "2016",

month = dec,

day = "1",

doi = "10.1002/malq.201400094",

language = "American English",

volume = "62",

pages = "530--546",

journal = "Mathematical Logic Quarterly",

issn = "0942-5616",

publisher = "Wiley-VCH Verlag",

number = "6",

}

TY - JOUR

T1 - Forcing a countable structure to belong to the ground model

AU - Kaplan, Itay

AU - Shelah, Saharon

N1 - 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

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85007441197&partnerID=8YFLogxK

U2 - 10.1002/malq.201400094

DO - 10.1002/malq.201400094

M3 - Article

AN - SCOPUS:85007441197

SN - 0942-5616

VL - 62

SP - 530

EP - 546

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

IS - 6

ER -

Forcing a countable structure to belong to the ground model

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this