The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this article, we introduce an operation for embeddings at the level of I0 and above, and prove that they generate an LD-algebra that can be quite different from the one implied by I3.

In this paper we introduce a new hierarchy of large cardinals between I3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathsf {I3}}}}$$\end{document} and I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathrm{\mathsf {I2}}}}$$\end{document}, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.

In this paper we introduce a new hierarchy of large cardinals between \ and \, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.