Timeline for Existence of surjection vs injection over $\sf ZF$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 3, 2022 at 20:37 | comment | added | Wojowu | @MartinVäth Hartogs' and Lindenbaum's theorems are precisely what my argument uses. I have essentially just expanded the arguments for their proofs. | |
| Jan 3, 2022 at 20:33 | comment | added | Martin Väth | Thank you. So the argument is essentially the same than proving/using Hartog's theorem (for injections) or the analogous Lindenbaum theorem (for surjections). These are not hard, but I was hoping for an essentially simpler proof. | |
| Jan 3, 2022 at 20:25 | comment | added | Wojowu | @MartinVäth For injections: for any $S$, you can form the set $A$ of all isomorphism classes of well-orders on subsets of $S$. By replacement, mapping each element of $A$ to its order type gives the class of ordinals which inject into $S$. For surjections, note that if $S$ surjects onto $\alpha$, then $\alpha$ injects into $P(S)$. | |
| Jan 3, 2022 at 20:21 | comment | added | Martin Väth | for any set $S$ the class of ordinals $\alpha$ such that $\alpha$ injects into $S$ (resp. $S$ surjects onto $\alpha$) is a set. Is there a simple argument to see this (especially for the "surjects" part)? | |
| Jan 1, 2022 at 21:03 | vote | accept | Dominic van der Zypen | ||
| Jan 1, 2022 at 15:32 | history | edited | Wojowu | CC BY-SA 4.0 |
added 88 characters in body
|
| Jan 1, 2022 at 15:26 | history | answered | Wojowu | CC BY-SA 4.0 |