Timeline for Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2014 at 0:49 | comment | added | Ameen | I got your point, thanks for your time being with me. | |
| Feb 4, 2014 at 0:36 | comment | added | Joel David Hamkins | The interval algebra is not complete, unless you complete it. But if you do complete it, then your argument about having the countable atomless Boolean algebra dense in it, also implies that it is the Cohen algebra. | |
| Feb 3, 2014 at 23:46 | comment | added | Ameen | Yes I mean that one. What I was thinking that the Boolean algebra generated by those intervals with rational end point would give us a countable atomless "dense" subalgebra $\mathcal Alg$, say. Thus, interval algebra would be the completion of $\mathcal Alg$, since all countable atomless subalgebras (have more that one element), and so is somorphic to Cohen algebra. | |
| Feb 3, 2014 at 23:32 | comment | added | Joel David Hamkins | Your interval algebra, if you mean just the Boolean algebra generated by those intervals, is not a complete Boolean algebra, but the Cohen algebra is usually taken as the complete Boolean algebra. But otherwise, they are related in that the Boolean completion of the interval algebra is isomorphic to the Cohen algebra. A condition in the Cohen algebra specifies a finite binary string, and each such string corresponds to a tiny interval in your interval algebra. | |
| Feb 3, 2014 at 23:04 | comment | added | Ameen | One more thing I would like to ask you if you do not mind. what is the relation between Cohen algebra and interval algebra. | |
| Feb 3, 2014 at 19:36 | vote | accept | Ameen | ||
| Feb 3, 2014 at 17:48 | comment | added | Ameen | Appreciate your ideas | |
| Feb 3, 2014 at 17:27 | comment | added | Joel David Hamkins | I expect that (2) will fail regardless of CH. | |
| Feb 3, 2014 at 17:21 | comment | added | Ameen | Thanks for the answer. I suspect that (1) might be false, but I could not bring any example (would be more helpful if you have an example in mind). But (2) might be true under the continuum hypothesis. What do you think? | |
| Feb 3, 2014 at 17:07 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |