0.999... Poll

What is your stance on this Paradox(0.999... repeating decimal and its value)
1.Transformatism(1 is exactly the same number as 0.999... written in different form)
2.Strong Convergism(1 is equal to 0.999... in all aspects)
3.Weak Convergism(0.999... converges to 1 at infinity)
4.Weak dualism(1 is equal to 0.999... practically, but is different in exact values)
5.Strong Dualism(1 is different and inequal number to 0.999...)
6.Separatism(1 and 0.999... belong to different number classes and cannot be compared)

>>14
Well, in the hyperreals you might define 0.999... to be \sum_{i=1}^n 9\cdot10^{-i} for some "infinitely large" hypernatural number n such that m<n for all m\in\mathbb{N}.  In this case the notation is ambiguous.

An algebraic approach is to simply define x<y on decimal expansions by looking at the first (from the left) digit in which x and y differ, so 0.\overline{9}<1.  However, if you work out the consequences of this things start to break down.  For example, if additive inverses exist then 9\cdot0.\overline{9}=9.\overline{9}-0.\overline{9}=9 which implies 0.\overline{9}=1.  We also don't have multiplicative inverses for all numbers (e.g., 3\cdot0.\overline{3}=0.\overline{9}, not 1!).  If you limit yourself to the nonnegative decimal expansions then addition and multiplication are well-defined, and you end up with an ordered semiring.  [For more detail, see "Is 0.999... = 1?", Fred Richman.]