[Version of October 31.]
I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.
- The consistency strength of is bounded above by a supercompact cardinal. See Foreman, Magidor, Shelah, Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2) 127 (1) (1988), 1-47.
- implies that there is a non-domestic premouse. This in turn implies that there is a sharp for an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. See Jensen, Schimmerling, Schindler, Steel, Stacking mice. The Journal of Symbolic Logic, to appear.
- is equiconsistent with a reflecting cardinal.
This is significantly stronger than in any forcing extension and that the existence of inner models of containing all the reals.
- is equiconsistent with an -reflecting cardinal .
- (for ) implies for all regular cardinals . For , this already implies the lower bound mentioned above for . For , this implies . See for example Schimmerling, Coherent sequences and threads. Adv. Math. 216 (1) (2007), 89-117.
- is consistent relative to a -reflecting cardinal . This follows from the results in Miyamoto, Localized reflecting cardinals and weak segments of . Preprint, 1996. The same holds with in place of .
- On the other hand, seems significantly stronger than . For example, implies that for every set there is an inner model that contains and has a strong cardinal, see Schindler, Bounded Martin’s Maximum and strong cardinals, in: Set theory. Centre de recerca Matematica, Barcelona 2003-4 (Bagaria, Todorcevic, eds.), Basel 2006, 401-406.
- An upper bound for the consistency strength of is Woodin cardinals. This is a result of Woodin, and it is buried somewhere in The Axiom of Determinacy, Forcing Axioms, and the Nonstationary ideal, De Gruyter, 1999, but I couldn’t find it this time; it probably follows from the arguments on on section 9.2. You can see a proof in Schindler’s talk The role of absoluteness and correctness, Barcelona, 2003 (pages 76-80). [Please let me know if you find the proof in Woodin’s book.]
Definition: For any cardinal , let denote restricted to forcings of size at most .
This notation is due to Woodin. I believe is a more useful notion, so one should probably call that one and this one , or something like that.
- is just Martin’s axiom for Cohen forcing. It is equiconsistent with .
- A bit more interestingly, is also equiconsistent with . I heard the rumor that implies , but I remain skeptical.
- Clearly, follows from , so it can be forced over and has consistency strength at most an -reflecting cardinal (also called a -indescribable cardinal). settles the size of the continuum, see Velickovic, Forcing axioms and stationary sets. Adv. Math. 94 (2) (1992), 256-284. This fragment has consistency strength at least a weakly compact (or -indescribable) cardinal.
- Foreman and Larson showed that implies , but their arguments do not even seem to adapt to . See Foreman, Larson, Small posets and the continuum. Unpublished note, 2004.
- Woodin has investigated in his book mentioned above, where he shows that it implies the existence of sharps for all subsets of (see section 9.5); this is the first step of an induction showing that implies . Later work of Steel and Zoble has extended this to show that it in fact implies , see Determinacy from strong reflection, preprint, September 16, 2008. Steel and Zoble conjecture that has consistency strength at least Woodin cardinals. Woodin also establishes (in section 9.2 of the book mentioned above) that is an upper bound for the consistency strength of . Recent work of Sargsyan shows that this determinacy hypothesis has strength below a Woodin limit of Woodin cardinals.
- implies .
- implies the lower bound mentioned above of an inner model with a proper class of Woodin cardinals and a proper class of strong cardinals.
Compare the following with the notion of indescribability defined above.
- and are equiconsistent with the existence of a such that is a -indescribable interval of cardinals. We also say that is -indescribable, and this notion relativizes to .
- More generally, the existence of a such that is -indescribable is an upper bound for the consistency strength of . For Neeman has shown that this is quite close to an equiconsistency. It is possible that the equiconsistency holds for all , and once inner model theory is developed at the appropriate level, we will be able to prove this. For , this upper bound falls beyond the upper bound for mentioned above.
- One can generalize the notions above to include -indescribability for all (see the Neeman, Schimmerling paper for details). A cardinal with a -indescribable interval is an upper bound for the consistency strength of .
Index of talks: