Pi is real number. We can enumerate any digit of Pi number.
Pi has infinite number of digits and it's enumeration in base 10 starts as follow:
What if we create the new number n from digits of Pi in reversed order? What if we will put every digit of Pi from float into natural part and from natural part into float. Reverse transformation works like mirror, the closest digits from float point will be mapped at the same distance into natural part and vice versa.
The new number n will looks like this:
...56295141.3 = n
The n will have infinite number of digits. We can enumerate any digit of this new number, but it has no first digit, similar as Pi which has no last digit.
Why it is allowed creation of numbers of infinite depth when we are going to smaller regions as Pi, but not allowed to create numbers of infinite size as n, where we are going to larger regions?
Number n can be written as infinite sum. How can we go beyond the real numbers, if we know that basic property of Real numbers is that if a,b are Real numbers then a+b is also Real number? We can write n as 0.3+1+4*10+1*100+5*1000... At each point we are adding real number to real number and there is no point where we can go outside of real numbers. We can even divide this sum into smaller sub sums and each of that smaller part also should be within real numbers.
If n is real number then n without floating point part can be natural number. If we could create natural number of infinite size, then we can show that Cantor's diagonal proof will fail, because in such case we will be able to create natural number of infinite size. If natural numbers can have infinite size then, we can create with the same diagonal manipulation on infinite size natural numbers as Cantor did with infinite depth Real numbers.
Is the n within Real number set?