The Isabelle theorem prover supports different logics, but its most developed seems to be Higher-Order Logic (HOL).  In this episode, I talk about the logic and approach of Isabelle/HOL, as far as I have understood them.  

I talk a bit more about the Agda proof assistant.

In this episode I talk a bit about the Agda proof assistant.

I talk about a couple good resources for learning Coq, the problem of too many ways to do things in type theory, and issues trying to explain and document a very complex language.

I discuss Coq, a widely used proof assistant based on a constructive type theory.  One episode definitely cannot do justice to the complexity of a tool like this -- but I take a first try at covering its features at a high level.

This is the start of Chapter 15, about interactive theorem provers (ITPs).  In this episode, I talk about the difference between fully automatic and interactive provers, and my plan to discuss and compare several different ITPs, in future episodes of this chapter.

In this episode, I outline the argument for why the proof-theoretic ordinal (in the sense of Rathjen, as presented last episode) is epsilon-0.  My explanation has something of a hole, in explaining how one would go about deriving induction for ordinals strictly less than epsilon-0 in Peano Arithmetic.  To help paper over this hole a little, I discuss a really nice recent exposition of encoding ordinals in Agda.

Ordinal analysis seeks to determine the strength of a logical theory by assigning an ordinal to it.  Which one?  In this episode I describe a definition of the proof-theoretic ordinal of a logical theory from a paper by proof theorist Michael Rathjen.  It is basically a measure of how strong an induction principle is derivable in the theory.  (The first parts of the paper are pretty accessible, but the rest gets hard, at least for me.)

Ordinal analysis is an important branch of proof theory, which seeks to compare, quantitatively, the strengths of different proof systems.  The quantities in question are ordinals, which extend the ordering character of natural numbers into the infinite.  In this episode, I discuss these ideas a bit further, and also review a little the ordinals up to epsilon 0.

Listen in while I have an actual insight on air!  After a week of head scratching, I figure out -- while chattering! -- what I hope is both a correct and intuitive example of multiplicative disjunction.