[FOM] a simple constructive example when AC is not true

[Dennis E. Hamilton]

From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
Kreinovich, Vladik
Sent: Tuesday, May 1, 2018 13:41
Subject: [FOM] a simple constructive example when AC is not true
 
In purely constructive approach, when existence is algorithmic and functions
are computable, by computing a computable real number x with accuracy 0.1,
we can efficiently check whether x > 0 or x < 1. 
[orcmid] 
 
To confirm my understanding of this setup, I need to be more specific.
 
Suppose that, for computable real, x, there is a computation that
approximates x by xa/10 where xa is an integer.  It is assured that xa/10 is
not greater than x and that x is less than (xa+1)/10.  This is by assurance
of x being a computable real and by construction of the algorithm for its
approximation.
 
One can generalize this formulation to consider a computable function for
xa =  xapprox(n) that produces integers such that xa/(10^n) is not greater
than x and x is less than (xa+1)/(10^n).  
 
It is trivially the case that (xa > 0 or xa < 1), and likewise for xa/(10^n)
and for x.  No efficiency is required.  Perhaps the better question is to
know whether x is in the closed interval [0,1] in which case it may be
necessary to get to an xapprox(n), for sufficiently large n, that settles
the question. 
 
Hence my confusion.
 
I am unclear how this, with needed repairs to my understanding about (x > 0
or x < 1), still tells us anything about AC with respect to the reals, since
xapprox(n) is not a real-valued function.  At best, it produces segments of
a Cauchy sequence of pairs of rationals, {xa[i]/(10^i), (xa[i]+1)/(10^i) }.
Note that xapprox(n) is not a function n reals and is not continuous in the
real-analysis sense.
 
What am I missing?
 
*	Dennis
 
 
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