BARC talks by S P Suresh

Monday and Tuesday, June 17 and 18 , 2024, S P Suresh, Professor of Computer Science at the Chennai Mathematical Institute, India, will give two lectures on "Introduction to Forcing".

Abstract / description of the lectures
In these lectures, we introduce independence proofs in set theory, particularly the method of forcing. This was a technique introduced by Cohen in 1963 to prove the independence of AC from ZF, and of CH from ZFC. ZF is the standard axiom system for set theory, formulated by Zermelo and Frankael. AC is the well-known Axiom of Choice, which is the assertion that every set can be well-ordered. CH is Cantor's Continuum Hypothesis, the assertion that every subset of the reals is of the same size as the reals or the same as of the naturals. 

Independence proofs are not unfamiliar to mainstream mathematicians. The most famous example is that Euclid's parallel postulate is independent of his other axioms, as there are geometries ("models") under which all of them are true, and geometries where the other axioms are true but the parallel postulate is false. These alternate geometries change the meaning of certain basic notions like lines or points. Similarly one can prove independence results for arithmetic as well, by changing what the symbol + means, for instance. All these independence proofs use the method of models. A model is a set with certain relations and functions defined on it. 

It is really challenging to construct alternative models for set theory, because sets are the basis using which we build other mathematical objects, but we seek to change the meaning of a set. Godel showed that, starting from a universe of sets that serves as a model for ZF, one can build an "inner model" (a subcollection of sets from the original universe) which serves as a model for ZF + AC + CH. But the inner model technique will not work to construct a model which falsifies AC or CH. Cohen in 1963 introduced the method of forcing for precisely this purpose. 

In two lectures, we aim to provide an overview of both Godel's and Cohen's constructions. We outline the logical and set theoretical framework needed for independence proofs, and provide the details of some of the constructions. 

The lectures are aimed at beginners, since the speaker himself is one! :-)

S P Suresh is a Professor of Computer Science at the Chennai Mathematical Institute, India. He completed his PhD in 2005 at the Institute of Mathematical Sciences, Chennai, India. His research interests are in formal verification, security theory, distributed computing, and functional programming. He is also interested in classical Indian systems of logic and philosophy.

Srikanth Srinivasan