Timeline for What is the consistency strength of $L(\mathbb{R})\models$ "$\omega_1$ has tree property"?
Current License: CC BY-SA 4.0
9 events
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| 6 hours ago | comment | added | Elliot Glazer | The definition Jech uses in his paper in which he proves $\omega_1$ can be weakly compact or measurable (each relative to such a large cardinal in ZFC) is the partition characterization: an uncountable well-ordered cardinal $\kappa$ is weakly compact if $\kappa \rightarrow (\kappa)^2_2.$ | |
| 7 hours ago | history | edited | new account | CC BY-SA 4.0 |
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| 7 hours ago | vote | accept | new account | ||
| 7 hours ago | comment | added | Joel David Hamkins | But the tree property alone is not equivalent to weak compactness even in ZFC, unless one also specifies that the cardinal is inaccessible. But $\omega_1$ is never inaccessible. | |
| 7 hours ago | answer | added | Farmer S | timeline score: 9 | |
| 7 hours ago | comment | added | new account | @JoelDavidHamkins Thanks for reminding me. I chose to use the definition via tree property and added the requirement that the underlying set should be $\kappa$, since otherwise I don't see how measurability implies weak compactness. Is it necessary? | |
| 7 hours ago | history | edited | new account | CC BY-SA 4.0 |
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| 7 hours ago | comment | added | Joel David Hamkins | There are over a dozen common characterizations of weak compactness in ZFC, but they are not all equivalent in ZF+DC, so could you let us know which version of weakly compact you want? | |
| 8 hours ago | history | asked | new account | CC BY-SA 4.0 |