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For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an arbitrarily large number of terms.

In other words, an

*completed*infinity makes no sense in the concept of limit.

However, the notion of infinity introduced by Georg Cantot does admit completed infinities. In Cantor's ideas, infinity does indeed exist an an actual quantity, and there are even a hierarchy of infinities.

How are these two seemingly opposed viewpoints reconciled? If we cannot define an actual sum of infinite terms because infinity has been replaced by the limit concept, how can we admit the actual infinities defined by Cantor?

I hope I am being clear. If not then let me know.