Hey /sci/ can someone tell me? I am really confused, and need to write a paper.
Why can't we rid maths of infinity?
It causes so many problems.
A collections of "things" where you can tell what belongs in it (but how many times is disregarded)
>A pair of "things" from S
Two things in S ordered by what's first and second
>A relation on a set S
A set of pairs of things in S
>An equivalence relation R on S
A relation on S that "generalizes equality". It's defined with the additional properties of
1. for any thing x in S, the pair (x, x) is always in R since a thing is always equivalent to itself (This is called reflexivity)
2. For any two things x and y in S, if (x, y) in R then (y, x) is too since if two things are equivalent then they are equivalent (This is called symmetry)
3. For any three things x, y, and z in S, if (x,y) and (y,z) are in R then (x,z) is too since if two things are equivalent to the same something else, then they are equivalent to each other (called transitivity)
The set of all numbers made up of a integer, a point, and a string of digits going on forever
>The real numbers
The set of decimals where equality is replaced by a special equivalence relation defined as "any two numbers are equivalent if they are the same number OR one nonzero number ends in all 0s and the other ends with all 9s and the last digits before the 9s is subtracted by 1"
Clearly we see that .9999...=1 BY DEFINITION of the real numbers.