Question

Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but not vice versa?

A hierarchy of coverings instead of infinities. Wouldn't that make those infinities more manageable in those terms?( yes I know topology can be expressed in set theory)

Answer 1

By $X$ covers $Y$ I assume you mean there exists a surjection $f:X \to Y$. The theory you're describing is exactly the same as the standard theory of cardinal numbers. In fact, if $X$ "covers" $Y$ and $Y$ "covers" $X$ then there is a bijection between $X$ and $Y$ . The proof is pretty and easy and is a good homework problem. You could also look it up in the beginning of any book that introduces the cardinals. Aside: I don't see what this has to do with topology. I also don't understand what you mean by "Dispensing with the notion of infinity..."

Answer 2

I don't quite know what you mean by "coverings of topology", but it is possible to formalize a notion of size for infinite sets which relies on the part-whole conception, rather than the bijective correspondence conception. These two views are mutually exclusive, in the sense that size for finite sets satisfies both properties, infinite sets can only support one of the two. But either choice can be made to work!

So, if you require the notion of size to equate two bijective sets, then the even numbers are equal in size to the natural numbers (this is the traditional Cantorian view). You could also take a mereological view, and say that one set is smaller than another if every element of the first set is a member of the second set. In this interpretation, the even numbers are smaller than the natural numbers.

A recent issue of the Review of Symbolic Logic had an article about these issues, including both some history of mathematics and more recent logical systems which formalize the mereological view. See Paolo Mancosu's "Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor's Theory of Infinite Number Inevitable?"