Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

Forcing lightface definable well-orders without the CGH

For any given uncountable cardinal $\kappa$ with $\kappa^{{<}\kappa}=\kappa$, we present a forcing that is $<\kappa$-directed closed, has the $\kappa^+$-c.c. and introduces a lightface definable well-order of $H(\kappa^+)$. We use this to define a global iteration that does this for all such $\kappa$ simultaneously and is capable of preserving the existence of many large cardinals in the universe

Laver and set theory

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