BibTeX
@MISC{Geuvers_onfixed,
author = {Herman Geuvers and Joep Verkoelen},
title = {On Fixed point and Looping Combinators in Type Theory},
year = {}
}
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Abstract
Abstract. The type theories λU and λU − are known to be logically inconsistent. For λU, this is known as Girard’s paradox [Gir72]; for λU − the inconsistency was proved by Coquand [Coq94]. It is also known that the inconsistency gives rise to a so called ”looping combinator”: a family of terms Ln such that Lnf is convertible with f(Ln+1f). It was unclear whether a fixed point combinator exists in these systems. Later, Hurkens [Hur95] has given a simpler version of the paradox in λU − , giving rise to an actual proof term that can be analyzed. In the present paper we analyze the proof of Hurkens and we study the looping combinator that arises from it: it is a real looping combinator (not a fixed point combinator) but in the Curry version of λU − it is a fixed-point combinator. We also analyze the possibility of typing a fixed point combinator in λU − and we prove that the Church and Turing fixed point combinators cannot be typed in λU −. 1
