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Re: distinguishability - in context, according to definitions
Posted:
Feb 16, 2013 6:55 AM
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I can't concentrate enough to understand the whole post, however :
Machines can't decide whether 1.0.... = 0.9999999... .
In general , machines can't decide the equality or inequality of real
numbers ,or infinite strings in general ,without the 0.(9) = 1
equivalence of real numbers.
This comes as a consequence that all computable functions are
continuous ,while equality is not.
http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html
Our language is countable , the real numbers are not . Thus we don't
work directly with "the plenum" , the real numbers as infinite strings
of digits . How could we? Who has time to read an infinite string?
What we do work with are the 'finite definitions' of these 'infinite
numbers' , for all the real numbers we can think about .Whether any
other kind of number is "real" ,other than those we can think about,
depends on your "orientation" in mathematics , though I affirm they're
not 'real' .
Now , these 'finite definitions' "subdue" the infinite numbers, making
their contents accessible to our tiny, finite minds . Thus , equality
becomes decidable , and 1 = 0.(9) while 1 not = 0.9999998(9) .
One question remains : Is anything lost when "replacing" these
"infinite objects" by their finite definitions?
The beautiful fact is that the objects of mathematics are analytic ,
not synthetic , thus nothing is lost in terms of meaning by saying
0.(9) instead of 0.99999.... and much is gained in terms of what we
could do with "the finite 0.(9)" , as opposed to "the infinite
0.9999..." .
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