In my lecture at the 2011 Congress on Logic, Methodology, and the Philosophy of Science in Nancy, France, I spoke on an additional axiom of set theory — the Proper Forcing Axiom — which has proved very successful in settling combinatorial problems concerning uncountable sets. Since I have already written a exposition on this subject [43], I have decided to address a broader question in this art...

Forcing is a technique that was discovered by Cohen in the mid 20th century, and it is particularly useful in proving independence results. The process involves adding new, generic elements to your universe and using them to prove a result. In many applications this is encapsulated by the addition of suitable axioms, often these are Forcing Axioms. One of the first such principles is known as M...

The notion of realizability algebra, which was introduced in [17, 18], is a tool to study the proof-program correspondence and to build new models of set theory, which we call realizability models of ZF. It is a variant of the well known notion of combinatory algebra, with a new instruction cc, and a new type for the environments. The sets of forcing conditions, in common use in set theory, are...

We give a very brief survey on ZFC theory (Zermelo-Fraenkel Set Theory) and we present an intuitive introduction to the method of forcing and some applications to the real line. Our exposition will be very informal, without any claim of completeness and rigour. The idea is just to give a very intuitive idea of what Set Theory and forcing-method are, and why they are interesting and useful from ...

Extender based forcings are studied with respect of adding branches to Aronszajn trees. We construct a model with no Aronszajn tree over אω+2 from the optimal assumptions. This answers a question of Friedman and Halilović [1]. The reader interested only in Friedman and Halilović question may skip the first section and go directly to the second. 1 No branches to κAronszajn trees. We deal here wi...

[1] Global simulations of sea salt and mineral dust aerosols are integrated into a previously developed unified general circulation model (GCM), the Goddard Institute for Space Studies (GISS) GCM II0, that simulates coupled tropospheric ozone-NOxhydrocarbon chemistry and sulfate, nitrate, ammonium, black carbon, primary organic carbon, and secondary organic carbon aerosols. The fully coupled ga...

It is assumed that a Kripke-Joyal semantics A = 〈C,Cov,F, 〉 has been defined for a first-order language L. To transform C into a Heyting algebra C on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of C. A pretopology Cov is defined on C using the pretopology on C. A sheaf F is made up of sections of F that obey fu...