Consider the set of all natural numbers and the set of all primes.
Since the set of all natural numbers less the set of primes is equal to the products or the elements of the subsets in the power set of the primes, then it is clear that mapping is injective, as all combinations are unique. This is by definition, as composite numbers are exactly those numbers that can the product of primes.
The cardinality of the set of natural numbers is infinite, but it is a much smaller infinity that that of real numbers, obviously, as there are an infinity of infinities (an infinite amount of irrational numbers between each rational, which are themselves infinite) in that set, but only a single infinity in that of the natural numbers.
Now consider: as the number progress to infinity, the likelihood of each number to be a product of some combination of each before it decreases to zero faster than the number line progresses. For example:
in {1,2}, there is a 50% chance of each being prime
in {1,2,...,10}, there is a 40% chance of each being prime
in {1,2,...,100}, there is a 25% chance of each being prime
in {1,2,...,1000}, there is a 16.8% chance of each being prime
in {1,2,...,10^10}, there is a 0.04% chance of each being prime
Thus, as infinity is approached, the number of primes approaches a finite number. This leads us to a contradiction, since any finite number of elements arranged in a any number of combinations will still only lead to a finite number of results.
Thus, since there cannot possibly be an infinite number of primes, there cannot be an infinite number of numbers.
The mere existence of infinity disproves infinity.
