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DanielLC comments on Artificial Addition - Less Wrong
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I read a book on the philosophy of set theory -- and I get lost right at the point where classical infinite thought was replaced by modern infinite thought. IIRC the problem was paradoxes based on infinite recursion (Zeno et. all) and finding mathematical foundations to satisfy calculus limits. Then something about Cantor, cardinality and some hand wavy 'infinite sets are real!'.
1.999... is just an infinite set summation of finite numbers 1 + 0.9 + 0.09 + ...
Now, how an infinite process on an infinite set can equal an integer is a problem I still grapple with. Classical theory said that this was nonsense since one would never finish the summation (if one were to begin). I tend to agree and I suppose one could say I see infinity as a verb and not a noun.
I suggest anyone who believes 1.999... === 2 really looks into what that means. The root of the argument isn't "What is the number between 1.999... and 2?" but rather "Can we say that 1.999... is a sensible theoretical concept?"
It was nonsense in classical theory. Infinite sum has its own separate definition.
There are times in modern mathematics that infinite numbers are used. This is not one of them.
I doubt I'm the best at explaining what limits are, so I won't bother. I may be able to tell you what they aren't. They give results similar to the intuitive idea of infinite numbers, but they don't do it in the most intuitively obvious way. They don't use infinite numbers. They use a certain property that at most one number will have in relation to a sequence. In the case of 1, 1.9, 1.99, ..., this number is two. In the case of 1, 0, 1, 0, ..., there is no such number, so the series is said not to converge.
No. The question is "Can we make a sensible theoretical way to interpret the numeral 1.999..., that approximately matches our intuitions?" It wasn't easy, but we managed it.