Timeline for Large cardinals without replacement
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2021 at 0:02 | vote | accept | Tim Campion | ||
| Feb 24, 2021 at 0:01 | comment | added | Dmytro Taranovsky | @TimCampion There is always a drop in strength, though it typically manifests only with contrived $B$. However, when the axiom is not $Σ_2$, there are natural examples: For example, ZC + "$κ$ is strong" does not prove that there is a $λ^{+ω}$-strong $λ$ in $V_κ$. | |
| Feb 23, 2021 at 23:37 | comment | added | Tim Campion | Thanks, there's a lot to dig into in this answer -- it really seems to cover all the bases! One lingering question -- for a large cardinal property $A(\kappa)$, $ZFC+\exists\kappa A(\kappa)$ typically proves that $V_\kappa$ models $ZFC+\exists \lambda B(\lambda)$ for certain weaker large cardinal properties $B$. And in your answer, you say that $ZF+\exists \kappa A(\kappa)$ typically proves that $V_\kappa$ models $ZFC+\exists \lambda C(\lambda)$ for certain weaker large cardinal properties $C$. Should I be thinking that $B = C$? Or is there rather sometimes a drop in strength from $B$ to $C$? | |
| Feb 23, 2021 at 23:13 | history | answered | Dmytro Taranovsky | CC BY-SA 4.0 |