Talk:Worldly cardinal

What if we change the V_κ in the definition to L_κ?

What is κ called if L_κ is a model of ZF? Does V_κ⊨ZF imply L_κ⊨ZF (as strongly inaccessible cardinals are weakly inaccessible)? 103.166.228.86 (talk) 01:24, 18 April 2024 (UTC)

If ⁠$V_{\kappa }\vDash {\mathsf {ZF}}$⁠, then ⁠$L_{\kappa }$⁠ is the constructible universe in the model ⁠$V_{\kappa }$⁠, so ⁠$L_{\kappa }\vDash {\mathsf {ZF}}$⁠. However, in general ⁠$L_{\alpha }\vDash {\mathsf {ZF}}$⁠ does not imply that ⁠$\alpha$⁠ is a cardinal, or even an ⁠$L$⁠-cardinal. I do not know if the class of ordinals ⁠$\{\alpha \mid L_{\alpha }\vDash {\mathsf {ZF}}\}$⁠ has been studied enough to have a name. Bbbbbbbbba (talk) 14:40, 16 July 2026 (UTC)