Very similar questions to this have been answered on the following threads, but they have not answered this particular question about the symmetry of the relation used for calculating the cardinality.
Here are some related threads:
The cardinality of the even numbers is half of the cardinality of the natural numbers?
Why is cardinality of set of even numbers = set of whole numbers?
What is larger -- the set of all positive even numbers, or the set of all positive integers?
Ratio of whole numbers to even numbers
The first link gets at my point, but does not answer my specific question on it, and I cannot ask my question with a short comment.
To set up and clarify my thoughts and question: First, my understanding is that mathematicians and logicians agree that the size of whole numbers and of even numbers are equal essentially because they are homeomorphic. Now, to be more precise, my understanding is that the set of even numbers is considered to be the same size as the set of whole numbers because When we map the whole numbers to even numbers by multiplying the whole number by 2 (as a bijective function as I understand it), we find that every whole number always has a corresponding even number paired with it.
However, it seems to me that for the 2 sets to really have the same cardinality, then then we'd need to get the same result when mapping the two sets together when flipping the sets and performing the same function for mapping This is because a) if the cardinality of set A (size 8) = the cardinality of set B (size 8), then the cardinality of set B (size 8) = cardinality of set A (size 8). b) Since we are dealing with infinite sets, we calculate their size using a mapping relation (also called a bijective function as I understand it) (like x * 2) instead of simply counting the elements in the set. therefore c) we need to also use the same function/relation to map set B to A in the same way we mapped A to B in order to be confident they really have the same size
In other words, for the sets to be the same size, wouldn't that relation need to be symmetric?
To say it yet another way: since we are "counting" the infinite set by this mapping technique (using a relation--i.e., * 2--to map them), then we would also need to "count" the mapping by the same relation in the other direction and find that there is also always a one to one mapping when you pick any number from either set.
If the mapping of the sets needs to be symmetric as I suppose, then this would quickly break down. Mapping from even numbers to whole (instead of whole to even), we proceed this way: 0 * 2 = 0; 0 has 2, 2 has 0 in both sets. So, we have a one to one mapping there. That's good. Then we multiply the next element in the even set. 2*2 = 4; so, there is a one to one mapping there from 2 in the even number set to 4 in the whole number set, but what about 1, 2, & 3 in the whole number set? They have no mapping. They have been skipped.
So, wouldn't this show that the infinite size of whole numbers is larger than the infinite size of even numbers? (Just affirming different sizes of infinity similar to how the Diagonal Proof does? https://dailycampus.com/2021/02/05/cantors-diagonal-argument-and-other-ways-to-unravel-infinities/ )
Am I making a incorrect assertion that the mapping technique for counting the sets' sizes need to be symmetric? If so, why?
