It sounds like he is looking at this from a CS point of view, which is odd because CS relies on much of the same logic as analysis, just with finite sets of machine numbers.
I mean it is true that calculation wise you can only do what can be computed in finite time with finite length. You can't actually do calculations with infinite nonrepeating digits, that is uncomputable, and you can only do approximations.
Further I still don't like the idea that putting a . and numbers to the right somehow generates infinitely more numbers.
en.wikipedia.org
The p-adic numbers show that it is conceivable to generate numbers like .....99999 with an infinite number of digits to the left.
The only reason natural numbers or integers cannot have infinite digits, is because they are defined as constructible, computable or potential infinities. That makes them always finite but since there is no greatest integer, or greatest natural number, if we actually allowed an actual infinity of time to pass, we'd have infinite digits.
Real numbers are defined as being allowed to use an actual infinity to produce the digits. All it is saying that an actual infinity is larger than a potential infinity. Or that computing in finite time will always be less than computing with infinite time.
But is it valid to use actual infinities? Do actual infinities even exist? If we try to compute or construct a real number, all constructible or computable real numbers would face a potential infinity and be no more numerous than the potential infinity of natural numbers or integers. The real numbers being able to use actual infinity as part of the definition, is indeed like using magic, it is something that simply cannot be constructed, computed, or perfomed by anyone in the real world.
Only approximations of real numbers are usable and computable. I think the idea of mathematics without real numbers, sounds very promising. Rather than approximating something, and simply using semantics to say we've grasped infinity, it is better to deal with the reality of the actual constrains on calculation that we have.