Timeline for Consistent hierarchy of axiomatic systems
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 17, 2011 at 21:17 | vote | accept | Andreas Thom | ||
| Mar 17, 2011 at 1:10 | comment | added | Jason | @Oliver: You probably meant *CON*(ZF + AD) implies CON(ZFC + "There exist infinitely many Woodin cardinals"). ZF + AD cannot prove this (assuming ZF + AD is consistent) or it would prove its own consistency since CON(ZFC + "There exist infinitely many Woodin cardinals") implies CON(ZF + AD). In a model where ZF + AD holds, its inner model $\text{HOD}^{L(\mathbb{R})}$ will model ZFC + "There exists infinitely many Woodin cardinals", but this is a proper class. You can for any $n \in \mathbb{N}$ prove the consistency of ZFC + "There exists $n$ Woodin cardinals" from ZF + AD b/c it holds in cuts. | |
| Feb 24, 2011 at 3:24 | comment | added | Oliver | ZF + AD proves the consistency of ZFC + "There exists a Woodin cardinal," not just ZF + "There exists a Woodin cardinal." In fact it proves the consistency of ZFC + "There exist infinitely many Woodin cardinals." | |
| Feb 24, 2011 at 3:12 | comment | added | Jason | I meant what I typed: ZFC + "There exists a Woodin cardinal". What does it seem to contradict so I can clarify what I mean? In general, we can have ZF models that think that a statement is impossible but still think it's relatively consistent with ZF (i.e., think that there's a set model of ZF where it's true). | |
| Feb 24, 2011 at 2:06 | comment | added | David Roberts♦ | >"In fact, ZF + AD proves the consistency of ZFC + "There exists a Woodin cardinal"" Is that right? Or should it be "ZF + "There exists a Woodin cardinal""? (Disclaimer - I'm not a set theorist, it just seems to contradict what you say a few lines later) | |
| Feb 24, 2011 at 1:47 | history | answered | Jason | CC BY-SA 2.5 |