Speaker:

W. Hugh Woodin
(Harvard University)

Title:

Ultimate L

Abstract:

The modern
mathematical study of infinity began in the period
187984 with a series of papers by Cantor that
defined the fundamental framework of the subject.
Within 40 years the key ZFC axioms for Set Theory
were in place and the stage was set for the
detailed development of transfinite mathematics,
or so it seemed. However, in a completely
unexpected development, Cohen showed in 1963 that
even the most basic problem of Set Theory, that of
Cantor's Continuum Hypothesis, was not solvable on
the basis of the ZFC axioms.
The 50 years since Cohen's announcement has seen a
vast development of Cohen's method and the
realization that the occurrence of unsolvable
problems is ubiquitous in Set Theory. This
arguably challenges the very conception of Cantor
on which Set Theory is based.
However, during this same period, the detailed
study of special cases of the Continuum Hypothesis
led to a remarkable success. This was the
discovery and validation of the determinacy axioms
for Second Order Number Theory. The resulting
theory is largely immune to Cohen’s method.
Almost 25 years before Cohen’s discovery of
forcing, Gödel discovered the Constructible
Universe of Sets and defined the axiom "V = L”
which is the axiom that asserts that every set is
constructible. This axiom implies the Continuum
Hypothesis and more importantly, Cohen’s method of
forcing cannot be used in the context of the axiom
"V = L”. However the axiom "V = L" is false since
it limits the fundamental nature of infinity. In
particular the axiom refutes (most) strong axioms
of infinity and it refutes the determinacy axioms
of Second Order Number Theory.
A key question emerges. Is there an “ultimate”
version of Gödel’s constructible universe yielding
an axiom "V = Ultimate L" which retains the power
of the axiom "V = L" for resolving questions like
that of the Continuum Hypothesis, which is also
immune against Cohen’s method of forcing, and yet
which does not refute strong axioms of infinity?
Such an axiom would necessarily provide the
generalization of the determinacy axioms of Second
Order Number Theory to the entire universe of
sets.
Until recently there seemed to be a number of
convincing arguments as to why no such ultimate L
can possibly exist. But the situation is now
changed.

Program:


14:0015:30 

16:0017:30 
Saturday
Feb. 17

1st Lecture 
Coffee Break

2nd Lecture

Sunday
Feb. 18

3rd
Lecture 
Coffee
Break 
Public
Lecture 

Location: 
Lecture Hall 1, IPM
Niavaran Building,
Niavaran Square, Tehran 
Registration:

The attendance is free, but
requires registration.
Please fill out the registration form
and
send it to gt@ipm.ir with
the subject "MCM".


