### Proof Theory of Martin-Löf Type Theory. An overview

Anton Setzer
2004 Mathématiques et sciences humaines
We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsistent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-Löf type theory with W-type and one microscopic universe containing only two finite sets
more » ... y two finite sets is carried out. Then we look at the analysis of Martin-Löf type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the concept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength. In his famous list of mathematical problems [Hilbert, 1900] , he posed as second problem to show the consistency of an axiomatisation of the real numbers developed by him. He argued that, if this axiomatisation is shown to be consistent, this would prove the mathematical existence of the concept of real numbers and of the continuum: consistency implies existence. He stated as well the main problem, namely that, if one shows the consistency of a theory for formalising mathematics in the same theory, one has not achieved anything: if the original theory is inconsistent, it proves everything, even its own consistency. So in order to achieve something, one has to do more: namely show the consistency using methods which are considered to be safe. According to Hilbert, finitary methods were to be considered to be safe. By finitary methods he considered finitary calculations, as we can carry them out on a piece of paper. Later his problem was generalised to what is now known as Hilbert's programme: to prove the consistency of axiom systems, in which certain parts of mathematics