## $$A_6^*$$ is equivalent to $$A_6$$ [WIP]

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…

## Extending Woodin’s definition of large cardinal axiom to all axioms

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 $$V_{\kappa_i}\prec_{\alpha_{i+1}} V_{\kappa_{i+1}}$$.

## On strong forms of extendibility

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…

## Large cardinals ordered by cardinality

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…