Speaker: Hugo Herbelin

Title: On the logical structure of some choice, maximality, bar induction and well-foundedness principles

Abstract:

The talk will be based on joint work with Nuria Brede and Jad Koleilat.

In a first step, we will show how the axiom of dependent choice and
the boolean prime ideal theorem can be expressed as particular
instances of a common general form of principle for choosing a
function from A to B subject to a set of finite approximations T
constraining the function. Then, A countable will capture the strength
of dependent choices, B two-valued the strength of the boolean prime
ideal theorem while T "prime" (in a sense to define) will give the
full axiom of choice. Dually, the contraposite of this principle will
characterize different strengths of bar induction or fan theorem.

To capture another class of principles equivalent to the axiom of
choice, we will develop in a second step a form of maximal choice
principle inspired from Teichmuller-Tukey's lemma, and of which Zorn's
lemma is an instance. Interestingly, its contraposite will then appear
to be a well-foundedness principle adapting to arbitrary cardinals a
principle from constructive mathematics known as Berger's update
induction.