Skip to main content
Bounty Ended with 500 reputation awarded by Elliot Glazer
added 19 characters in body
Source Link
Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k

Basically, the forcing is an Easton-support productiteration of adding a Cohen subset to each regular cardinal. This forces both the GCH and the ground axiom, and the closure features accommodate the master condition arguments that allow supercompact and higher large cardinals to be preserved, including rank-to-rank.

Basically, the forcing is an Easton-support product of adding a Cohen subset to each regular cardinal. This forces the ground axiom, and the closure features accommodate the master condition arguments that allow supercompact and higher large cardinals to be preserved, including rank-to-rank.

Basically, the forcing is an Easton-support iteration of adding a Cohen subset to each regular cardinal. This forces both the GCH and the ground axiom, and the closure features accommodate the master condition arguments that allow supercompact and higher large cardinals to be preserved, including rank-to-rank.

deleted 2 characters in body
Source Link
Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k
added 54 characters in body
Source Link
Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k

GCH. The standard forcing of the GCH is an Easton-support iteration that simply iteratively forces the next instance directly, and this forcing has very nice structural features that will enable to preserve most all of the usual large cardinals by the usual lifting arguments. The usual master condition arguments, the same arguments used in the Laver preparation, show that this forcing will preserve all supercompact cardinals, and one can use the same methods to preserve larger large cardinals. For example, every measurable cardinal, every strong cardinal, every supercompact cardinal, almost every partiallpartially supercompact cardinal, every rank-to-rank cardinal, and so forth, is preserved by the canonical forcing of the GCH. So the GCH is relatively consistent with most all of the usually considered large cardinals.

Ground Axiom. The ground axiom, introduced by myself and Jonas Reitz, is similarly consistent with large cardinals. This follows by one of the main dissertation resultresults of Jonas Reitz, who was my student at the time.

Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh, The ground axiom is consistent with (V \neq \text{HOD}), Proc. Am. Math. Soc. 136, No. 8, 2943-2949 (2008). ZBL1145.03029.

GCH. The standard forcing of the GCH is an Easton-support iteration that simply iteratively forces the next instance directly, and this forcing has very nice structural features that will enable to preserve most all of the usual large cardinals by the usual lifting arguments. The usual master condition arguments, the same arguments used in the Laver preparation, show that this forcing will preserve all supercompact cardinals, and one can use the same methods to preserve larger large cardinals. For example, every measurable cardinal, every strong cardinal, every supercompact cardinal, almost every partiall supercompact cardinal, every rank-to-rank cardinal, and so forth, is preserved by the canonical forcing of the GCH. So the GCH is relatively consistent with most all of the usually considered large cardinals.

Ground Axiom. The ground axiom is similarly consistent with large cardinals. This follows by the main dissertation result Jonas Reitz, who was my student at the time.

Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh, The ground axiom is consistent with (V \neq \text{HOD}), Proc. Am. Math. Soc. 136, No. 8, 2943-2949 (2008). ZBL1145.03029.

GCH. The standard forcing of the GCH is an Easton-support iteration that simply iteratively forces the next instance directly, and this forcing has very nice structural features that will enable to preserve most all of the usual large cardinals by the usual lifting arguments. The usual master condition arguments, the same arguments used in the Laver preparation, show that this forcing will preserve all supercompact cardinals, and one can use the same methods to preserve larger large cardinals. For example, every measurable cardinal, every strong cardinal, every supercompact cardinal, almost every partially supercompact cardinal, every rank-to-rank cardinal, and so forth, is preserved by the canonical forcing of the GCH. So the GCH is relatively consistent with most all of the usually considered large cardinals.

Ground Axiom. The ground axiom, introduced by myself and Jonas Reitz, is similarly consistent with large cardinals. This follows by one of the main dissertation results of Jonas Reitz, who was my student at the time.

added 1044 characters in body
Source Link
Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k
Loading
Source Link
Joel David Hamkins
  • 236.5k
  • 44
  • 777
  • 1.4k
Loading