FOM: Intuitionistic cuts
Harvey Friedman
friedman at math.ohio-state.edu
Sat Mar 14 10:26:31 EST 1998
Pratt 10:02AM 3/14/98 writes:
>I am not sure what to make of my difficulty in expressing the bound of
>two on the number of blocks intuitionistically. Can it be done, or is
>there some other intrinsic problem with the cut-based definition of the
>reals that Brouwer overlooked?
Yes. An intuitionistically correct cut-based formulation is given as
follows. We have two sets A,B of rational numbers, such that
i) A,B are disjoint and each have an element;
ii) (x in A & y < x) implies y in A;
iii) (x in B & y > x) implies y in B;
iv) if x < y then x in A or y in B.
Here x,y are rationals, and A,B are sets in the sense of Brouwer's theory
of species. And we define (A,B) < (C,D) if and only if B,C meet. One
proves, e.g., that this ordering is isomorphic to the (quasi) ordering of
Cauchy sequences with prescribed rate of convergence.
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