Timeline for Cardinality of connected manifolds
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 21, 2011 at 20:09 | comment | added | Ali Enayat | @Juris: right, I see your point. By the way, is there a similar proof [using elementary submodels] of Arhangel'ski's theorem about the maximum cardinality of first countable Lindelof Haussdorf spaces? | |
| Jun 18, 2011 at 20:09 | comment | added | Juris Steprans | @Ali: The phrase "elementary submodel of a large fragment of set theory" does sweep a bit under the rug. I had in mind simply, for any manifold taking a submodel of $V_\kappa$ where $\kappa$ is larger than he cardinality of the manifold. Of course, this also the strategy for proving reflection. | |
| Jun 17, 2011 at 16:46 | comment | added | Ali Enayat | @Juris: Nice solution! Technically speaking, Lowenheim-Skolem cannot be invoked unless we are working in Kelley-Morse theory of classes (where there is a truth-definition for $(V,\in))$; but what "saves the day" and makes your proof implementable in $ZF$, is the $ZF$-Reflection Theorem. | |
| Jun 17, 2011 at 0:34 | comment | added | Juris Steprans | Yes, this follows from the Lowenheim-Skolem Theorem. | |
| Jun 17, 2011 at 0:06 | comment | added | Mariano Suárez-Álvarez | It is obvious that such an $\mathfrak M$ exists? | |
| Jun 16, 2011 at 22:54 | history | answered | Juris Steprans | CC BY-SA 3.0 |