A Mathmetics Blog
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 \(V_{\kappa_i}\prec_{\alpha_{i+1}} V_{\kappa_{i+1}}\).
1. Abstract The large cardinal program studies the properties of large cardinals; cardinals with large cardinal properties. Many large cardinal properties have been discovered: Inaccessibility; Mahloness, weak compactness, indescribability, partition properties, \(0^\sharp\), measurability, \(0^\dagger\), and the hierarchy of huge cardinals. The last collection of properties is particularly interesting. The hierarchy of huge cardinals consists of…
Based on my answer to this question on Math Overflow: https://mathoverflow.net/questions/250978/large-cardinals-ordered-by-cardinality-of-least-instance/333418#333418 1. The standard ordering of large cardinals. The gaps between large cardinals are much more complex then just \(LC_1\lt LC_2\) can express. There are two main types of large cardinals. Limit cardinals and embedding cardinals. A limit cardinals is created by taking a class…