Timeline for Negation of cardinal characteristics
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 7, 2022 at 14:36 | comment | added | Ioannis Souldatos | @Goldstern I like your characterization of $\aleph_1$, but you can not change the value of this "characteristic". | |
| Oct 31, 2022 at 16:48 | comment | added | Goldstern | Is $\aleph_1$ a meaningful characteristic? This is the largest cardinal of a subset of the reals which carries a well-order all initial segments of which are Borel. | |
| Oct 31, 2022 at 16:45 | comment | added | Goldstern | So you don't consider the cardinal characteristic $\mathfrak c= 2^{\aleph_0}$ meaningful? | |
| Jan 23, 2020 at 13:52 | comment | added | Will Brian | The results about the Cohen and random model can be found in chapter 8 of Bartoszynski and Judah's book Set Theory: on the structure of the real line. For the random real model, it follows from Theorem 8.2.8 that there are uncountable strong measure zero sets, and it is Theorem 8.2.11 that every strong measure zero set has size $\leq \aleph_1$. | |
| Jan 23, 2020 at 0:31 | comment | added | Ioannis Souldatos | @WillBrian. Can you give me a reference for the consistency results? I am more interested in the random reals case, where $\mathfrak{s}_+$ is strictly between $\aleph_0$ and $2^{\aleph_0}$. | |
| Jan 22, 2020 at 19:27 | comment | added | Ioannis Souldatos | @WillBrian. Then this answers the question! Do you mind writhing it out so I can give you some credit :) | |
| Jan 22, 2020 at 19:06 | comment | added | Will Brian | It is a theorem that $\mathfrak{s}_+$ is a cardinal between $\aleph_0$ and $2^{\aleph_0}$ (inclusive). It is consistent that it is $\aleph_0$ (this happens in Laver's model -- google "Borel conjecture" to read more about it), that it is uncountable but less than $2^{\aleph_0}$ (this happens in the random real model), and that it is equal to $2^{\aleph_0}$, even when CH fails (this happens in the Cohen model). | |
| Jan 22, 2020 at 18:55 | comment | added | Ioannis Souldatos | @WillBrian. This seems a good candidate, but is it consistent that $\mathfrak{s}_+$ is a cardinal between $\aleph_0$ and $2^{\aleph_0}$? | |
| Jan 22, 2020 at 18:38 | comment | added | Ioannis Souldatos | @WillBrian: I have to see what is the strong-measure-zero subsets of R. | |
| Jan 22, 2020 at 14:17 | comment | added | Will Brian | What about the supremum of the cardinalities of strong-measure-zero subsets of $\mathbb R$? I don't know if you'd consider it a "cardinal characteristic" or not, especially since it is consistent that this cardinal is countable, but otherwise it fits the bill. There's a question about it here: mathoverflow.net/questions/281024/the-strong-measure-number/…. | |
| Jan 22, 2020 at 3:51 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Jan 21, 2020 at 22:21 | history | asked | Ioannis Souldatos | CC BY-SA 4.0 |