“Potential infinite” and “actual infinite” are the very foundation for present classical infinite related science system and really play important roles every where. But if we can not know what the “infinite things” exactly are, how can we behave in our practical mathematical operations?

Dx are and the union of n are conditionally infinite in composition, depending on what it is you are trying to prove.  I'd say because of infinity's abstract conceptually broad nature it does remain prevalent in calculus as well as with Cardinality as a legitimate tool.    In the harmonic series 1/k where k is an element of the reals we have divergence as the sum tends to infinity but if we have the 1/k in form of a alternating series the sum converges at the ln(2).  But if we take the partial sum  of  the alternating series we don't know if it diverges?  

Dear Dr. Daniel O'Hearn , thank you.

In fact, we have many definitions for the infinite related number forms of the Un--->0 in mathematics historically. But, the problem is so far no matter what definition is (standard analysis or its equivalent non-standard analysis or…), no one can tell “how many numbers each bigger than any positive constants (such as Un’ >1000000000000000)” can be produced from the Un--->0 items in Harmonic Series by the “brackets-placing rule" of present infinite related theory and skills.

It seems those historical definitions for the infinite related number forms of the Un--->0 are language games, not for solving mathematical problems-------that is the reason for “Why infinite small could not be included in mathematical definition for infinite in present classical infinite related science system?”.

Best regards

Geng