Month: November 2019

In there paper Strong Axioms of Infinity and Elementary Embedding, Robert Solovay, William Reinhardt, and Akihiro Kanamori introduced the axioms \(A_1\)-\(A_7\), measuring the gap between hugeness and extendibility. \(A_1(\kappa)\): \(\kappa\) is huge. \(A_2(\kappa)\): There is some \(j: V_\alpha\prec V_\beta\) with critical point \(\kappa\) such that \(j(\kappa)\le\alpha\). \(A_3(\kappa)\): \(\kappa\) is almost huge; i.e. there is some…

Let \(\phi(\kappa)\) be a local large cardinal axiom if and only if it is of the form \(\exists\kappa_0…\kappa_n(\phi(\kappa,\kappa_0…))\land\forall i\le n(\kappa_i\in C^{(\alpha_i)})\), such that \(V_{\kappa_i}\prec_{\alpha_{i+1}} V_{\kappa_{i+1}}\).