Timeline for Correspondence between functions on a set and "states" on its power set
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2010 at 21:55 | vote | accept | tortortor | ||
| May 12, 2010 at 23:45 | comment | added | Joel David Hamkins | Yes, that's it. I should have said that the existence of a counterexample to your correspondence is equivalent to the existence of a real-valued measurable cardinal. | |
| May 12, 2010 at 23:31 | comment | added | tortortor | Thank you very much for the fast answer. If I understand correctly: What I called a "state" can be obtained from any real valued measure. Such a measure may exist for P(X) (with X continuum) but only in violation of the Continuum Hypothesis etc. But even if we assume the existence of such measures, the correspondence to the functions (in my sense) on X can no longer hold, as all such measures must assign zero to singletons? | |
| May 12, 2010 at 18:41 | comment | added | Joel David Hamkins | Yes, I agree, and I was writing the explanation in the meantime. | |
| May 12, 2010 at 18:40 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 1620 characters in body; added 423 characters in body
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| May 12, 2010 at 18:29 | comment | added | Gerald Edgar | But of course the OP is asking about real-valued measurable cardinals. Even if their existence may be equiconsistent with 2-valued measurable cardinals, still real-valued measurable cardinals need not be "large" ... In fact the most interesting universe is one where $\mathbb{R}$ itself admits a real-valued measure on its power set. (Ulam showed that $\aleph_1$ is not real-valued measurable.) | |
| May 12, 2010 at 18:13 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |