Torsion-free abelian groups are consistently aΔ12-complete

Saharon Shelah; Douglas Ulrich

doi:10.4064/fm673-12-2018

Torsion-free abelian groups are consistently aΔ12-complete

Saharon Shelah, Douglas Ulrich

Einstein Institute of Mathematics

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

Abstract

Let TFAG be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZFC- + "R(ω) exists", then TFAG is aΔ12- complete; in particular, this is consistent with ZFC. We define the α-ary Schröder- Bernstein property, and show that TFAG fails the α-ary Schröder-Bernstein property for every α < R (ω). We leave open whether or not TFAG can have the R(ω)-ary Schröder- Bernstein property; if it did, then it would not be aΔ12 -complete, and hence not Borel complete.

Original language	English
Pages (from-to)	275-297
Number of pages	23
Journal	Fundamenta Mathematicae
Volume	247
Issue number	3
DOIs	https://doi.org/10.4064/fm673-12-2018
State	Published - 2019

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2019.

Keywords

Borel complexity
Torsion-free abelian groups

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10.4064/fm673-12-2018

Cite this

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abstract = "Let TFAG be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZFC- + {"}R(ω) exists{"}, then TFAG is aΔ12- complete; in particular, this is consistent with ZFC. We define the α-ary Schr{\"o}der- Bernstein property, and show that TFAG fails the α-ary Schr{\"o}der-Bernstein property for every α < R (ω). We leave open whether or not TFAG can have the R(ω)-ary Schr{\"o}der- Bernstein property; if it did, then it would not be aΔ12 -complete, and hence not Borel complete.",

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AU - Ulrich, Douglas

N1 - Publisher Copyright: © Instytut Matematyczny PAN, 2019.

PY - 2019

Y1 - 2019

N2 - Let TFAG be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZFC- + "R(ω) exists", then TFAG is aΔ12- complete; in particular, this is consistent with ZFC. We define the α-ary Schröder- Bernstein property, and show that TFAG fails the α-ary Schröder-Bernstein property for every α < R (ω). We leave open whether or not TFAG can have the R(ω)-ary Schröder- Bernstein property; if it did, then it would not be aΔ12 -complete, and hence not Borel complete.

AB - Let TFAG be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZFC- + "R(ω) exists", then TFAG is aΔ12- complete; in particular, this is consistent with ZFC. We define the α-ary Schröder- Bernstein property, and show that TFAG fails the α-ary Schröder-Bernstein property for every α < R (ω). We leave open whether or not TFAG can have the R(ω)-ary Schröder- Bernstein property; if it did, then it would not be aΔ12 -complete, and hence not Borel complete.

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KW - Torsion-free abelian groups

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JO - Fundamenta Mathematicae

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IS - 3

ER -

Torsion-free abelian groups are consistently aΔ12-complete

Abstract

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Keywords

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