On Mar 16, 6:03 pm, Earle Jones <earle.jo...@comcast.net> wrote: > In article <virgil-16CD4A.02232808022...@BIGNEWS.USENETMONSTER.COM>, > > > > > > > > > > Virgil <vir...@ligriv.com> wrote: > > In article > > <8422f6ea-4891-4c77-b538-b3a192619...@z4g2000vbz.googlegroups.com>, > > JT <jonas.thornv...@gmail.com> wrote: > > > > On 8 Feb, 09:22, Virgil <vir...@ligriv.com> wrote: > > > > In article <firstname.lastname@example.org>, > > > > > spermato...@yahoo.com wrote: > > > > > simply > > > > > 0.9999.... is a non-finite number/ > > > > > For the number represented by 0.999..., which is clearly greater than > > > > zero, to be infinite, it would have to at least be greater than 1 as > > > > well, but is not! > > > > > If you mean that 0.999... represents an infinitely long numeral, that is > > > > something quite different. > > > > > If you cannot distinguish between a numeral and the number it > > > > represents, you are too ignorant to be posting to sci.math. > > > > -- > > > > It is the same concepts Virgil you should now > > > I do not know how to 'now' concepts. > > > > that 1/inf of the same > > > size as N/inf, > > > That depends on whether N is a member of the set of naturals numbers or > > is the set of natual numbers itself. > > * > If 0.999... is not equal to 1, > then there is a number between 0.999... and 1 > > Please write it here_______________ > > Thanks, > > earle > *
There's a saying "you can't see the forest, for the trees." So, the reals are defined as (the least) complete ordered field. These are the real numbers of the line, the points of Hardy, scalar values, elements of the linear continuum. Then, another consideration of the linear continuum is at is is started as the sweep, of the drawing, of the line, from zero through one. These are also the line, a gapless/ complete linear continuum of values. Then, .999... can have two meanings, that of the dual representation of 1.0 as a Dedekind/Cauchy/ Eudoxus expansion, or as a notation for that which occurs in the sweep after all other elements and right before 1.0, in the total natural ordering of the points of the real line from zero through one, as . 000...001 would follow zero: in a notation.
So, _defining_ .999... as equal to one, follows from the definition of the expansion, _defining_ .999... as less than one, follows from the definition of the drawing. They're not interchangeable: instead simply reflecting the differing perspectives of the structure: of the line, from in it (as a set, of each point), and on it (as the sweep, for each point).
Those careful definitions are consistent, in the correct context, simply not interchangeable, but: the notation reflects both the intuitive ideas.