We investigate the compatibility of $$I_0$$ with various combinatorial principles at $$\lambda ^+$$, which include the existence of $$\lambda ^+$$ -Aronszajn trees, square principles at $$\lambda $$, the existence of good scales at $$\lambda $$, stationary reflections for subsets of $$\lambda ^{+}$$, diamond principles at $$\lambda $$ and the singular cardinal hypothesis at $$\lambda $$. We also discuss whether these principles can hold in $$L(V_{\lambda +1})$$.

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.

In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:Vλ+1≺Vλ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j : V_{\lambda+1} {\prec} V_{\lambda+1}}$$\end{document} with the failure of GCH (...) at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}. (shrink)

In this paper we introduce a generic large cardinal akin to, together with the consequences of being such a generic large cardinal. In this case is Jónsson, and in a choiceless inner model many properties hold that are in contrast with pcf theory in.

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding Lα → Lα. Using these inverse limit reflection embeddings directly and by broadening the collection of U-representable sets, we prove structural results of L under the assumption that there exists an elementary embedding j : L → L. As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ Vλ+1, thus focusing the (...) study of L generalizations on L. (shrink)