arXiv:1007.3018math.HO
When is .999... less than 1?
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Abstract
We examine alternative interpretations of the symbol described as nought, point, nine recurring. Is "an infinite number of 9s" merely a figure of speech? How are such alternative interpretations related to infinite cardinalities? How are they expressed in Lightstone's "semicolon" notation? Is it possible to choose a canonical alternative interpretation? Should unital evaluation of the symbol .999 . . . be inculcated in a pre-limit teaching environment? The problem of the unital evaluation is hereby examined from the pre-R, pre-lim viewpoint of the student.
Of course that argument only applies when talking about the standard real number system. http://arxiv.org/pdf/1007.3018.pdf?origin=publication_detail http://arxiv.org/pdf/0811.0164.pdf ...of course you can make a pretty good case that uncountable, uncomputable, unnameable…
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