Timeline for How the cardinalities of $\mathcal H^*_x$ and $\mathcal P(x)$ compare?
Current License: CC BY-SA 4.0
11 events
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| May 15 at 13:06 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 15 at 13:03 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, Yes, but the question is about when $x$ is a Dedekind infinite. Also, I need the relation to be $\leq$ and not just $<$. As for the transitive closure definition being itself smaller, I've given this alternative to compare with the main alternative. I need some freedom in comparisons between the powersets and the hereditarily size sets, that's why I'm asking about both versions. I'll modify the notation a little bit as not to be confused with the ordinary one. | |
| May 15 at 12:39 | comment | added | Joel David Hamkins | You say "every element of the transitive closure of $\{y\}$ is small", but this is exactly making the mistake I mentioned. The usual definition would say that the transitive closure itself is small. Also, you are using $\leq$ instead of $<$, which means that if ZFC holds, your terminology does not match the established terminology. For example, your notation has no way to refer to the hereditarily finite sets. | |
| May 15 at 5:11 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 15 at 4:43 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, I've edited to clarify. | |
| May 15 at 4:42 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 14 at 23:34 | comment | added | Joel David Hamkins | Sorry, you don't need GCH in my earlier comment. It is true in ZFC. | |
| May 14 at 23:29 | comment | added | Joel David Hamkins | Another issue is that even in ZFC we don't define $H_\kappa$ as the collection of sets that are hereditarily of size less than $\kappa$, if what this means is that they have size less than $\kappa$, and their elements also, and the elements of those sets, etc. This gives the wrong result when $\kappa$ is singular. One normally defines $H_\kappa$ as the collection of sets whose transitive closures have size less than $\kappa$. I'm not sure what "subnumerous" means, but in the ZF context all talk of cardinality is more subtle and complicated, so it might help to say more exactly what is meant. | |
| May 14 at 23:26 | comment | added | Joel David Hamkins | Wouldn't it be more natural to want to compare $H_\kappa$ with $P_\kappa(\kappa)$, which is the set of subsets of $\kappa$ of size less than $\kappa$. In ZFC+GCH, these have the same size for all infinite regular $\kappa$. | |
| May 14 at 23:12 | comment | added | Joel David Hamkins | When $x$ is $\omega$, then $H_\omega$ is countably infinite, but $P(\omega)$ is strictly larger, even in ZF. | |
| May 14 at 22:49 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |