Timeline for bijections and order types
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2018 at 3:00 | comment | added | Alex Kruckman | @AndreasBlass Ah great, thanks for supplying the argument. | |
| Nov 8, 2018 at 2:58 | comment | added | Andreas Blass | I think you can proceed similarly when $\kappa$ has countable cofinality, using $\kappa+\omega_1$ instead of $\kappa+\omega$. The preimage $X$ of the final $\omega_1$ must have order-type $\leq\omega_1$ by the condition in the problem, but it can't be $<\omega_1$ by cardinality. So $X$ has order-type exactly $\omega_1$, and then $X$ can't be cofinal in $\kappa$. Then pick a larger element $\alpha<\kappa$ and notice that $X\cup\{\alpha\}$ has order-type $\omega_1+1$ while its image has order-type only $\omega_1$, just as in your proof. | |
| Nov 8, 2018 at 2:13 | vote | accept | Monroe Eskew | ||
| Nov 8, 2018 at 2:11 | history | answered | Alex Kruckman | CC BY-SA 4.0 |