ℵ₀, the infinity that represents the cardinality of natural numbers, would not be "any infinity" in the set of hyperreal numbers. You have a fundamental misunderstanding of the concept of infinitea if you cannot point to a specific number that "bounds the natural numbers" because that number is ℵ₀ and can be pointed to. It is the only countable infinity. Bring in irrationals and now it is uncountable, because there are an infinite number of numbers between 1 and 2. You can never reach 2 if you counted every number between 1 and 2. The cardinality of irrational numbers is ℵ1, a distinctly different and larger infinity than ℵ₀.

Sets like naturals and rationals may have the same cardinality, but they are not functionally the same. Cardinality is just one attribute they share. The powers of 10 cannot be analytically continued to -1/12 like the natural numbers can. Therefore they are functionally different.