Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B.
Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraically that's clearly Limit(0->inifinity) of X/2 which is infinity.
PS: What makes Cantor's diagonalization interesting is you can repeat it recursively an infinite number of times. This is more obvious in base 2.
The reals, on the other hand, cannot be placed in a bijection with the natural numbers, and there are therefore "more" reals than naturals (i.e. there is an injection from the naturals to the reals, but not from the reals to the naturals -- any function from reals to naturals must have some pair x ≠ y with f(x) = f(y)).
Is the set of Real Numbers larger, smaller, or the same size as the set of points in a finite 2d object? Can you setup a bijection in either direction?
If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number.
Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up, there will be a number you would miss.
The primes and rationals have the same size (cardinality) as the natural numbers, namely countably infinite. See https://en.wikipedia.org/wiki/Countable_set#Formal_overview_...
There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.
Diagonalization isn't showing that a number in set A isn't in set B - that's obviously true for reals and integers, but it's also true for rationals and integers. It's showing that there does not exist a mapping from B to A where there's an element in B for each element in A.
We're obviously not using the same definition of "size". I generally think in terms of cardinality, what are you thinking of?
If every element in set A is in set B, and there are elements in set A left over it's larger because that's what larger means. {A,B} < {A,B,C}
There are countable and uncountable infinite set's. All countable set's have a bijection with N. However, there are more than two sizes of infinite sets. Real numbers < Imaginary numbers.
