Logic in Computer Science, Symposium on (2007)

Wroclaw, Poland

July 10, 2007 to July 14, 2007

ISSN: 1043-6871

ISBN: 0-7695-2908-9

pp: 23-34

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/LICS.2007.7

Wojciech Moczydlowski , Cornell University, USA

ABSTRACT

Set theories are traditionally based on first-order logic. We show that in a constructive setting, basing a set theory on a dependent logic yields many benefits. To this end, we introduce a dependent impredicative constructive set theory which we call IZF_D. Using realizability, we prove that the underlying lambda calculus weakly normalizes, thus enabling program extraction from IZF_D proofs. We also show that IZF_D can interpret IZF with Collection. By a wellknown result of Friedman, this establishes IZF_D as a remarkably strong theory, with proof-theoretical power equal to that of ZFC. We further demonstrate that IZF_D provides a natural framework to interpret first-order definitions, thus removing a longstanding barrier to implementing constructive set theories. Finally, we prove that IZF_D extended with excluded middle is consistent, thus paving the way to using our framework in the classical setting as well.

INDEX TERMS

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CITATION

Wojciech Moczydlowski,
"A Dependent Set Theory",

*Logic in Computer Science, Symposium on*, vol. 00, no. , pp. 23-34, 2007, doi:10.1109/LICS.2007.7