[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

        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), 
 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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