[tomtomtom1]

>"but you're no longer talking the same language as the rest of the world"

yes, in the standard real numbers 1 = 0.999.., but people have dealt with numbers like "pi" and "sqrt(2)" before the standard real numbers were defined.

Hence the question, if we define such a system such as 0.333... != 1/3. what are the consequences?

by 0.3333... I mean a countably infinite sequence of 3s.

[atq2119]

I think an important distinction is that in those "old days", people were largely working in what we now know to be subsets of real numbers, and the same conclusion applies there.

If you want to go to supersets of real numbers, you may be interested in https://en.wikipedia.org/wiki/Surreal_number

[Scarblac]

If 0.333... != 1/3, then they are different numbers and the expression 1/3 - 0.333... must have some value different from zero. What is that value?