[FOM] Construction of the Reals and a Paradigm
A. Mani
a_mani_sc_gs at yahoo.co.in
Wed Nov 16 06:35:30 EST 2005
Hello,
Starting from Q , we can construct R by defining a tolerance on the
former. [(x,y)\,\in\,T iff |x-y| \leq 1, the set of blocks along with the new
operations is isomorphic to R. ] This was proved in
Czedli, G "Factor Lattices by Tolerances" Acta Sci Math (Szeged), 44 (1982),
35-42.
A tolerance is a reflexive and symmetric relation on a set. It is a
'compatible tolerance' on an algebra if it is also compatible with the
operations of the algebra.
Interpret the rationals as a totally ordered lattice.
A 'Block' of the tolerance T is a maximal subset B of Q s.t. B^{2} \subseteq
T. Blocks are to tolerances what classes are to equivalences. Tolerances can
be fully defined by their blocks (Chajda', et. al 1976, Gratzer, et.al 1989
Arch. Math (Brno)25 ). For lattices, the general
representation of compatible tolerances by blocks in an algebra gets
considerably simpler and longer. In fact it is quite convenient to write Q|T
to mean the set of blocks of T .
Thm. If L is a lattice and T a tolerance on it, then L|T is also a lattice
with the operations on L|T being defined by
A v B = {a v b : a\in A, b\in B} and dually. ]
This can be interpreted as 'by describing the inexact' sufficiently well we
can get to exact models... Because
(x, y)\in T means "x is approximable by y" (in some other senses too apart
from the usual mathematical one.)
It is easy to redefine the tolerance without using the |.| function.
What is the minimal set theory (whichever) required in the construction ?
Is there an intuitionistic reformulation of this construction ?
Tolerances have been best studied in universal algebra, but using them more
directly in mathematical foundations is another thing. Though I have
developed generalised algebraic semantics of logics based on tolerances.
A. Mani
Member, Cal. Math. Soc
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