## Proof Theory of Martin-Lof Type Theory -- An (2004)

Venue: | Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99 |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Setzer04prooftheory,

author = {Overview Anton Setzer and Anton Setzer},

title = {Proof Theory of Martin-Lof Type Theory -- An},

booktitle = {Mathematiques et Sciences Humaines, 42 année, n o 165:59 – 99},

year = {2004},

pages = {42--59}

}

### OpenURL

### Abstract

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-Lof type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-Lof 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.

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