Can Natural Numbers Be Taken To Infinity?

This is not really a claim of mine so much as a question.

There is a video going viral out there that explains the very real mathematical result that the sum of all natural numbers is equal to -(1/12). It's an interesting video, and factually true. But I like the "second proof and additional footge" video better. View it here.

This is not my area of expertise, but after watching both videos and reading some more about it, I am comfortable with the conclusion, except for one small detail: I am not sure it makes sense to take an infinite sum of the set of Natural Numbers.

If I had to prove this, I think it would require the second Peano Axiom: Every natural number "a" has a natural number successor "a + 1".

The videos and their underlying mathematics are definitely true; I am not making the claim that there is anything incorrect about these proofs, except the nomenclature. What I am actually saying is that the infinite sum of the set of all Real Numbers that look like Natural Numbers is -(1/12). There are also set theory methods of constructing a set that looks like the Natural Numbers. But I think the phrase "Natural Numbers" has been traditionally used to denote simple digits used for counting. It wouldn't make sense to take that set to infinity; it might not be mathematically wrong, but it undermines the spirit of that set of numbers.

This claim may be irrelevant semantics, or it may be plainly wrong. My point is simply to suggest that there is nothing "natural" about taking the Natural Numbers to infinity, which is not itself a Natural Number.

Perhaps this is a pointless post. I am entitled to a few of those, from time to time. 

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