@MISC{Pavlovic99oncoalgebra, author = {D. Pavlovic and V. Pratt}, title = {On Coalgebra of Real Numbers}, year = {1999} }

Share

OpenURL

Abstract

We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor X \Delta !, ordinal product with !. This makes an attractive analogy with the definition of the ordinal ! itself as the initial algebra of the functor 1; X , prepend unity, with both definitions made in the category of posets. The variants 1; (X \Delta !), X o \Delta !, and 1; (X o \Delta !) yield respectively Cantor space (surplus rationals), Baire space (no rationals), and again the continuum as their final coalgebras. 1 Introduction Coinduction has only relatively recently been recognized as a genuine logical principle [2]. Before that, it was introduced and used mostly in the semantics of concurrency [13]. It has by now been presented from many different angles: [1,8,12,16--18], to name just a few contributors. Why would so foundational a principle wait for the late 20th century to be discovered? In [14,16] the idea was put forward that coinduction is new only by nam...