Abstract
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.
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Greenberg was partially supported by a Rutherford Discovery Fellowship and a Lady Davis visiting professorship.
Shelah’s research partially supported by the German-Israeli Foundation for scientific research and development, grant no.: I-706054.6/2001; and by the Israel Science Foundation, grant no.: ISF 1838/19. Paper 832 on Shelah’s list.
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Greenberg, N., Shelah, S. Many forcing axioms for all regular uncountable cardinals. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2570-0
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DOI: https://doi.org/10.1007/s11856-023-2570-0