Maximal almost disjoint families, determinacy, and forcing
Research output: Contribution to journal › Journal article › Research › peer-review
Standard
Maximal almost disjoint families, determinacy, and forcing. / Bakke Haga, Karen; Schrittesser, David; Törnquist, Asger.
In: Journal of Mathematical Logic, Vol. 22, No. 1, 2150026, 2022, p. 1-42.Research output: Contribution to journal › Journal article › Research › peer-review
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Maximal almost disjoint families, determinacy, and forcing
AU - Bakke Haga, Karen
AU - Schrittesser, David
AU - Törnquist, Asger
PY - 2022
Y1 - 2022
N2 - We study the notion of -MAD families where is a Borel ideal on ω. We show that if is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite -MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective -MAD families; and under the full Axiom of Determinacy + V = L(a) or under AD+ there are no infinite-mad families. Similar results are obtained in Solovay's model. These results apply in particular to the ideal Fin, which corresponds to the classical notion of MAD families, as well as to the ideal Fin. The proofs combine ideas from invariant descriptive set theory and forcing.
AB - We study the notion of -MAD families where is a Borel ideal on ω. We show that if is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite -MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective -MAD families; and under the full Axiom of Determinacy + V = L(a) or under AD+ there are no infinite-mad families. Similar results are obtained in Solovay's model. These results apply in particular to the ideal Fin, which corresponds to the classical notion of MAD families, as well as to the ideal Fin. The proofs combine ideas from invariant descriptive set theory and forcing.
KW - Borel ideals
KW - Definability
KW - determinacy
KW - Fubini product
KW - Mathias forcing
KW - maximal almost disjoint families
UR - http://www.scopus.com/inward/record.url?scp=85105997429&partnerID=8YFLogxK
U2 - 10.1142/S0219061321500264
DO - 10.1142/S0219061321500264
M3 - Journal article
AN - SCOPUS:85105997429
VL - 22
SP - 1
EP - 42
JO - Journal of Mathematical Logic
JF - Journal of Mathematical Logic
SN - 0219-0613
IS - 1
M1 - 2150026
ER -
ID: 276955168