The infinity symbol is defined loosely as "a very large number" and is not treated as a value. You can't multiply it, you can't divide it, you can't add other infinite values to it. If you create a new symbol which does behave naturally you get a much better mathematics which doesn't suffer from the metaphysical rubbish like mapping (bijection) a set of order O(N) to a set of order O(N²).

Here is a more useful set of tools to work with infinity: http://lesliegreen.byethost3.com/articles/new_maths.pdf