Timeline for Is there an abstract theory of club sets and stationary sets?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 21, 2019 at 12:57 | vote | accept | Joseph Van Name | ||
| Mar 21, 2019 at 12:35 | review | Close votes | |||
| Mar 21, 2019 at 17:02 | |||||
| Mar 17, 2019 at 3:45 | comment | added | Joseph Van Name | The notion of stationarity in Example 3 was first introduced by Donna Carr. This notion of stationarity is mentioned in Chapter 25 of Kanamori's The Higher Infinite. | |
| Mar 13, 2019 at 6:25 | answer | added | Ali Enayat | timeline score: 4 | |
| Mar 12, 2019 at 7:08 | comment | added | Monroe Eskew | Where does example 3 come from? Something about supercompact measures? | |
| Mar 11, 2019 at 15:56 | comment | added | Joseph Van Name | In fact, if $C_{F}$ denotes the sets closed under the function $F$, then the closure systems of the form $C_{F}$ are precisely the algebraic closure systems where if $R\subseteq X$ is infinite and $S$ is the $C_{F}$-closure of $R$, then $|R|=|S|$. In other words, the closures are all algebraic closures cannot boost infinite cardinalities. This is similar to topological closure since in a regular space, we have $|\overline{R}|\leq 2^{2^{|R|}}.$ | |
| Mar 11, 2019 at 14:51 | comment | added | Joseph Van Name | @MonroeEskew. If $F:X^{<\omega}\rightarrow X$, then the collection of sets $C$ closed under $F$ is what is known as an algebraic closure system. So one could rephrase Woodin's notion to saying that $Z$ is stationary precisely when for each suitable algebraic closure system $C$ on $X$, the set $C\cap Z$ is nonempty. Perhaps one could further abstract Woodin's notion of stationarity by allowing a broader class of closure systems besides just the algebraic closure systems. | |
| Mar 11, 2019 at 13:02 | comment | added | Monroe Eskew | It is well-known that the notion of club in Example 2 is equivalent to that in Example 1, when restricted to the club $\kappa \subseteq P_\kappa(\kappa)$. I don't know about example 3. Is it non-equivalent to the usual notion? | |
| Mar 11, 2019 at 12:59 | comment | added | Monroe Eskew | The following notion is due to Woodin. Let $Z$ be any set and let $X = \bigcup Z$. We say $Z$ is stationary if for every function $F : X^{<\omega} \to X$, there is $z \in Z$ such that $z$ is closed under $F$. It is a theorem of Kueker that the notion of club for $P_\kappa(\lambda)$ mentioned in Example 2 is equivalent to: "$C$ is in the club filter iff there is a function $F : \lambda^{<\omega} \to \lambda$ such that $C$ contains the set of $z$ such that $z \cap \kappa \in \kappa$ and $z$ is closed under $F$." | |
| Mar 11, 2019 at 3:56 | comment | added | Mohammad Golshani | You may also look at the answer given at Stationarity and Fodor's lemma for a (nice) poset? | |
| Mar 10, 2019 at 20:21 | comment | added | Joseph Van Name | In the above paper, the authors ask the question as to what the correct notion of a club set will be in a filter space, so it seems like the notion of a filter space may need to be modified to encapsulate club sets. | |
| Mar 10, 2019 at 19:37 | comment | added | Joseph Van Name | About 45 minutes after posting this question, I just recalled this paper that attempts to classify large cardinal axioms using filter spaces and these filter spaces give an abstract notion of stationarity. Has such an abstract theory of filter spaces been developed further by anyone? sciencedirect.com/science/article/pii/0168007289900092 | |
| Mar 10, 2019 at 18:44 | history | asked | Joseph Van Name | CC BY-SA 4.0 |