Timeline for Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2023 at 14:39 | comment | added | Holo | About the question of isomorphic $\aleph_1$-dense subsets, Sierpinski showed that it is possible that there exists $2^{\aleph_0}$-isomorphism classes of $\aleph_1$-dense subsets of reals (Sierpinski assumed CH, but one should get similar result with the failure of CH by adding Cohen reals), on the other hand Baumgartner showed that PFA implies that all $\aleph_1$-dense subsets of real are isomorphic | |
| Feb 10, 2023 at 2:54 | comment | added | Noah Schweber | @bof "reflecting" is just the dual of "preserving" - $f(a)\le f(b)\rightarrow a\le b$. Meanwhile, I used injections because then it doesn't matter whether you're using "order-preserving" in the strict or weak sense. | |
| Feb 10, 2023 at 1:40 | comment | added | bof | What does "reflecting" mean? And what's the significance of injectivity here? Non-injective order-preserving maps would be just as good here, right? | |
| Feb 10, 2023 at 1:22 | vote | accept | Dominic van der Zypen | ||
| Feb 9, 2023 at 3:46 | comment | added | Noah Schweber | @bof Good point, added! | |
| Feb 9, 2023 at 3:46 | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 507 characters in body
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| Feb 9, 2023 at 3:35 | comment | added | bof | Also, since there are order-preserving maps ($x\lt y\implies f(x)\lt f(y)$) in both directions, $\mathbb R\to\mathcal P(\omega)$ and $\mathcal P(\omega)\to\mathbb R$, the linear orders embeddable in $\mathcal P(\omega)$ are just the same as the linear orders embeddable in $\mathbb R$. | |
| Feb 8, 2023 at 21:08 | history | answered | Noah Schweber | CC BY-SA 4.0 |