Can I prove that there are 3 times more even numbers than natural numbers by assigning each even number to a the next three natural numbers starting 0

Question

Recently I have been thinking about infinity and more specifically, sets of an infinite size, or an infinite amount of elements in them. And I found this concept of a bijection where if each element in one infinite set can correspond to another element in another infinite set, without any overlap, then the two sets have the same 'cardinality' or the same number of elements within them. Using this method you can also prove that there are the same number of reals from 0-1 as there are 0-2. But can I prove that there are three times more even numbers than natural numbers by assigning three natural numbers to every even number? What is the flaw in this logic?

Answer 1

Even if your logic was right, you would show that there are three times as many natural numbers as even numbers.

Anyway, the cardinality of infinite sets is defined such that if there exists a bijection between a set $X$ and the natural numbers, $\mathbf{N}$, $|{X}| = |\mathbf{N}|$. Since there clearly exists such a bijection (e.g. to each even number, assign its half), the sets have equal cardinality (denoted $\aleph_0$). From then on, any other mapping is irrelevant, and with infinities, comparisons such as "three times as many" do not make sense anyway, as the basic arithmetic operations for infinite cardinalities do not coincide with what we are used to with natural numbers.