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Topic: distinguishability - in context, according to definitions
Replies: 43   Last Post: Feb 22, 2013 10:04 AM

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 dan.ms.chaos@gmail.com Posts: 409 Registered: 3/1/08
Re: distinguishability - in context, according to definitions
Posted: Feb 16, 2013 6:55 AM

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..." .

Date Subject Author
2/10/13 fom
2/10/13 J. Antonio Perez M.
2/10/13 fom
2/11/13 Shmuel (Seymour J.) Metz
2/11/13 fom
2/14/13 Shmuel (Seymour J.) Metz
2/14/13 fom
2/14/13 fom
2/15/13 fom
2/15/13 Shmuel (Seymour J.) Metz
2/16/13 fom
2/17/13 Shmuel (Seymour J.) Metz
2/19/13 fom
2/21/13 Shmuel (Seymour J.) Metz
2/15/13 fom
2/15/13 fom
2/14/13 fom
2/17/13 Shmuel (Seymour J.) Metz
2/17/13 fom
2/17/13 Barb Knox
2/18/13 fom
2/19/13 Shmuel (Seymour J.) Metz
2/19/13 fom
2/21/13 Shmuel (Seymour J.) Metz
2/19/13 Shmuel (Seymour J.) Metz
2/19/13 fom
2/21/13 Shmuel (Seymour J.) Metz
2/21/13 fom
2/22/13 Shmuel (Seymour J.) Metz
2/15/13 fom
2/17/13 Shmuel (Seymour J.) Metz
2/17/13 fom
2/19/13 Shmuel (Seymour J.) Metz
2/16/13 dan.ms.chaos@gmail.com
2/16/13 fom
2/17/13 dan.ms.chaos@gmail.com
2/17/13 fom
2/17/13 dan.ms.chaos@gmail.com
2/18/13 Shmuel (Seymour J.) Metz
2/20/13 fom
2/21/13 Shmuel (Seymour J.) Metz
2/16/13 fom
2/19/13 Shmuel (Seymour J.) Metz
2/19/13 fom