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That's because, in terms of cardinality, we say \(\infty \times 5 = \infty\).

Or, to be more specific \(\aleph_0 \times 5 = \aleph_0\).

Cardinality is a property of sets that can be understood as the size of the set.

The sum of the elements in a set have no relation to cardinality.

So what's the concept I'm looking for if it's not cardinality?

No, cardinality is the concept you're looking for when it concerns the whole even/odd numbers

You might better find an error in your proof if you formalize it more