The language that math speaks about
a ≤ x ≤ b is based on collection of 0(), end exactly because
a,
x or
b are no more than 0(), thay can't be 1(), which exists simultaneously at AND beyond the location of any given 0().
This fact is notated exactly by ≠ of 1(0(
a)≠0(
x)≠0(
b)) expression or by 1(0(x)≠0

) expression, as well.
Your enunciation and the strange symbolism are the major ingredients in the recipe for the bowl of goulash you've been trying to serve.
No one claims that 0() is 1(), as no one claims that 0=1. If you organize the members of the set of real numbers in the ascending order, the result resembles a line. You claim that there are gaps in that set (collection), but you can't show that there is an interval [a,b] for which there is no real number x
i.
Let x
i be a real number -- a member of an ordered interval. What is the next real number?
Obviously, it is x
i+1. But you claim that there are cases where x
i has a successor x
i+2. I suggested to you to hit the ordered set of positive integers and find the gaps in there, before trying to demolish
R.
Let's see if you can get beyond shuffling 0()s and 1()s around: What is the immediate successor of
√2? If the immediate successor is
t, how do you find out that there should be a real number
s right between
√2 and
t? This is the same as spotting a gap in 1,2,3_5,6... You can spot the gap only when you know how the missing number looks like, and if you know that, then the number exists. There is no way that there is any gap there, coz if i say let A be the set of positive integers, then I mean a set with no gaps. If I meant otherwise, I would included the provision in the definition.
So, what is the immediate successor of
√2?