Timeline for $L_{\omega_1,\omega}$ sentence with many automorphism in $\aleph_0$ and few automorphism in $\aleph_\omega$
Current License: CC BY-SA 3.0
7 events
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| Jun 3, 2013 at 18:20 | comment | added | Ioannis Souldatos | @Joel: Thank you for the answer. So, it turns out that we can impose more restrictive conditions (first-order vs infinitary $\phi$; 1 vs $\le\aleph_\omega$ automorphisms) and still get a positive result. I was a little curious about Shelah's result. I sent him an email asking about it. If he gives me a reference, I will post it here. | |
| Jun 3, 2013 at 18:10 | vote | accept | Ioannis Souldatos | ||
| May 29, 2013 at 23:40 | comment | added | Joel David Hamkins | That is a common convention; the competing convention is that DLO means just "dense linear order", which has four completions, depending on how you settle the endpoint question. | |
| May 29, 2013 at 23:33 | comment | added | Noah Schweber | Yeah, I think EDLO is better than DLO for this sentence. But don't Marker/others tend to use DLO for "dense linear order without endpoints?" | |
| May 29, 2013 at 23:30 | comment | added | Joel David Hamkins | Yes, that's right. (But I would say EDLO, since with just DLO, you haven't settled the endpoint question.) | |
| May 29, 2013 at 23:27 | comment | added | Noah Schweber | One last step in this answer: the formula DLO in this example is in fact a complete infinitary (well, finitary, but finitary$\subseteq$infinitary) formula, since it is $\aleph_0$-categorical; see Corollary 2.12 at the end of homepages.math.uic.edu/~marker/inf.pdf. | |
| May 29, 2013 at 22:51 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |