A Cook's Tour of the Finitary Non-Well-Founded Sets

Details A Cook's Tour of the Finitary Non-Well-Founded Sets A Cook's Tour of the Finitary Non-Well-Founded Sets

2011-11-30

arxiv.org/abs/1111.7148v2

Computer Science

Logic in Computer Science

This paper is a write-up of a lecture originally given in 1988. It appeared in the Festschrift for Dov Gabbay, We Will Show Th

Scientific paper

We give multiple descriptions of a topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets. This universe is characterized as a metric completion of the hereditarily finite sets; as a Stone space arising as the solution of a functorial fixed-point equation involving the Vietoris construction; as the Stone dual of the free modal algebra; and as the subspace of maximal elements of a domain equation involving the Plotkin (or convex) powerdomain. These results illustrate the methods developed in the author's 'Domain theory in logical form', and related literature, and have been taken up in recent work on topological coalgebras. The set-theoretic universe of finitary sets also supports an interesting form of set theory. It contains non-well founded sets and a universal set; and is closed under positive versions of the usual axioms of set theory.

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