TY - JOUR
T1 - Many forcing axioms for all regular uncountable cardinals
AU - Greenberg, Noam
AU - Shelah, Saharon
N1 - Publisher Copyright:
© The Author(s) 2023.
PY - 2024/6
Y1 - 2024/6
N2 - A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and having a strong forcing axiom of higher order than usual. Instead of “every suitable forcing notion of size λ has a sufficiently generic filter” we shall say “for every suitable method of producing notions of forcing based on a given stationary set, there is such a suitable stationary set S and sufficiently generic filters for the notion of forcing attached to S”. Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
AB - A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and having a strong forcing axiom of higher order than usual. Instead of “every suitable forcing notion of size λ has a sufficiently generic filter” we shall say “for every suitable method of producing notions of forcing based on a given stationary set, there is such a suitable stationary set S and sufficiently generic filters for the notion of forcing attached to S”. Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
UR - http://www.scopus.com/inward/record.url?scp=85176747164&partnerID=8YFLogxK
U2 - 10.1007/s11856-023-2570-0
DO - 10.1007/s11856-023-2570-0
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AN - SCOPUS:85176747164
SN - 0021-2172
VL - 261
SP - 127
EP - 170
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -