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We analyze the notion of guessing model, a way to assign combinatorialproperties to arbitrary regular cardinals. Guessing models can be used, incombination with inaccessibility, to characterize various large cardinalsaxioms, ranging from supercompactness to rank-to-rank embeddings. The majorityof these large cardinals properties can be defined in terms of suitableelementary embeddings j\colon V_\gamma \to V_\lambda. One key observation isthat such embeddings are uniquely determined by the image structures j [V_\gamma ]\prec V_\lambda. These structures will be the prototypes guessingmodels. We shall show, using guessing models M, how to prove for the ordinal\kappa_M=j_M (\crit(j_M)) (where \pi_M is the transitive collapse of M and j_Mis its inverse) many of the combinatorial properties that we can prove for thecardinal j(\crit(j)) using the structure j[V_\gamma]\prec V_{j(\gamma)}.\kappa_M will always be a regular cardinal, but consistently can be asuccessor. Guessing models M with \kappa_M=\aleph_2 exist assuming the properforcing axiom PFA. By means of these models we shall introduce a new structuralproperty of models of PFA: the existence of a "Laver function" f : \aleph_2 \toH_{\aleph_2} sharing the same features of the usual Laver functions f:\kappa\to H_\kappa provided by a supercompact cardinal \kappa. Furtherapplications of our analysis will be proofs of the singular cardinal hypothesisand of the failure of the square principle assuming the existence of guessingmodels. In particular the failure of square shows that the existence ofguessing models is a very strong assumption in terms of large cardinalstrength.

Guessing models and generalized Laver diamond