Timeline for Is there a statement equivalent to a sentence admitting $(\alpha^{+n},\alpha)$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 25, 2023 at 18:17 | history | edited | LSpice | CC BY-SA 4.0 |
Unicode -> TeX
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| Jul 10, 2012 at 17:31 | vote | accept | Ioannis Souldatos | ||
| May 23, 2012 at 22:32 | answer | added | David Milovich | timeline score: 5 | |
| May 15, 2012 at 21:08 | comment | added | Joel David Hamkins | See en.wikipedia.org/wiki/Morass_(set_theory) for some information and a list of references. | |
| May 15, 2012 at 19:11 | comment | added | Ioannis Souldatos | @ Joel David Hamkins: I think it is my turn to ask for a definition :). What is a gap-n morass? | |
| May 15, 2012 at 15:11 | comment | added | Joel David Hamkins | I wonder if using a morass instead of a Kurepa tree might be a source of examples, since here one uses a system of small structures to approximate a large structure. The gap-n morass case is similar to your question, with a gap-n difference in the cardinals. | |
| May 15, 2012 at 14:31 | history | edited | Ioannis Souldatos | CC BY-SA 3.0 |
Added definition of admits $(\kappa,\lambda)$.
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| May 15, 2012 at 14:25 | comment | added | Ioannis Souldatos | One more comment: The tree described in (1) is an Aronszajn tree. The family described in (2) is a Kurepa family. They are both very well-known objects among set-theorists. So, maybe there another object that can capture $(\alpha^{+3},\alpha)$, $(\alpha^{+4},\alpha)$ and so on. | |
| May 15, 2012 at 14:22 | comment | added | Ioannis Souldatos | @Joel David Hamkins:Yes! A sentence $\sigma$ in a language with a unary predicate $P$ admits $(\kappa,\lambda)$, if $\sigma$ has a model $M$ such that $|M|=\kappa$ and $|P^M|=\lambda$. Of course, $\lambda\le\kappa$. So, not only we need a model of a specific size, but we need the predicate $P$ to have a specific size too. | |
| May 15, 2012 at 13:13 | comment | added | Joel David Hamkins | Can you remind us what it means to "admit $(\kappa,\lambda)$"? Does it mean that the language has a predicate and you are asking for a model of size $\kappa$, where the predicate has size $\lambda$? | |
| May 15, 2012 at 12:21 | history | edited | Ioannis Souldatos |
edited tags
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| May 15, 2012 at 12:14 | comment | added | Ioannis Souldatos | For the problem we are working on, we assume GCH. Nevertheless, I would be interested to hear what you have in mind. | |
| May 15, 2012 at 4:36 | comment | added | Péter Komjáth | Does $2^\alpha\geq\alpha^{+n}$ qualify? | |
| May 15, 2012 at 1:30 | history | asked | Ioannis Souldatos | CC BY-SA 3.0 |