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..." .