Timeline for Basic cardinal arithmetic without choice
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2023 at 2:32 | comment | added | Hanul Jeon | To be honest, I do not know. | |
| Apr 28, 2023 at 2:14 | comment | added | new account | @HanulJeon That paper and its references mention Borel equivalence relations a lot. Is it true that under determinacy axioms $E$ reduces to $E'$ iff there is an injective set map from $X/E$ to $Y/E'$? If not, how exactly is Borel cardinality related to true cardinality? | |
| Apr 28, 2023 at 1:59 | comment | added | Noah Schweber | @newaccount You almost never get universality in such a sense when you're working with definitions in a logic as strong as first-order logic; consider definitions like "$0$ if $\mathsf{AC}$ and $1$ if $\neg \mathsf{AC}$." No model of $\mathsf{ZF}$ can simultaneously separate this from both $0$ and $1$. | |
| Apr 27, 2023 at 4:49 | comment | added | new account | @AsafKaragila But my last question does concern arbitrary $\mathsf{ZF}$ models. An attempt to formalize it: fix a bunch of definitions of cardinalities, modulo provable equinumerosity in $\mathsf{ZF}$; for each model $M$ there is an homomorphism from those definitions to actual cardinalities in $M$; is there an $M$ that is universal, namely all other homomorphisms factor through it | |
| Apr 27, 2023 at 4:41 | comment | added | new account | @AsafKaragila If we only consider, e.g., finitely many parameter-free definable sets (instead of arbitrary sets), can the semi rng they generate be simply described? At first I was hoping that the structure generated by $\mathbb{R}$ and $\omega_1$ is "the symmetric algebra of the monoid $\{\emptyset,\mathbb{R},\omega_1,\mathbb{R}+\omega_1\}$", but that's far from true. But it still seems interesting if given two sets $A,B$ there is an algorithm to figure out whether they generate a symmetric algebra. And I'm more interested in determinacy model than arbitrary $\mathsf{ZF}$ model | |
| Apr 27, 2023 at 3:50 | comment | added | Asaf Karagila♦ | You can have models of ZF in which every partial order emebds into the cardinals, and so there always large sets of pairwise incomparable cardinals. It seems to me that those will have a very crazy structure of arithmetic. But as a whole I'm not sure what is your actual question. | |
| Apr 27, 2023 at 2:10 | comment | added | Hanul Jeon | Woodin's paper The cardinals below $|[\omega_1]^{<\omega_1}|$ seems relevant to your question. | |
| S Apr 27, 2023 at 1:53 | review | First questions | |||
| Apr 27, 2023 at 8:28 | |||||
| S Apr 27, 2023 at 1:53 | history | asked | new account | CC BY-SA 4.0 |