/sci/ - Science & Math

>>11496310
it's easy to point out the flaw in the logic right here
>0 -> infinity = divergence
>0 -> 1 = convergence
because there is the assumption that there are infinite numbers between 0 and 1, we come to the conclusion that
>"0 -> 1" and "0 -> infinity" contain the same amount of expansion
meaning
>divergence = convergence
>convergence = divergence
and then the value of either convergence or divergence needing to be terms that are used in any capacity disintegrates.

What exactly do you mean by "expansion"? If you mean the cardinality of the set of real numbers between 0 and 1 equals that of the set of all positive real numbers, then yes, that's one of the weird, fucky results you get by playing with the concept of infinity. It's harder to show with the real numbers, but pic related is a neat way of thinking about the countability of rational numbers, aka there are "as many" rational numbers as there are integers.

In any case, though, I think you're misapplying the definition of a limit. 0 -> infinity means that, for a given positive epsilon, we can choose integer N such that n > N implies the nth term in the sequence is greater than epsilon; similarly, 0 -> 1 means that the normed difference between the nth term and 1 is less than epsilon. Unbounded growth and bounded approach are very different, even if we use the same notation.

In any case, your concept of "expansion" is far too nebulous to do any real math with. One might even say its "value" as a term that is "used in any capacity disintegrates".

The thing is, the limit definitions above might be conceptually fucky, but they're *useful*. They're how we prove and how, over the past hundreds of years, we've steadily built upon the concepts of calculus. I don't care how weird they might seem (which is really more a question of familiarity, anyways); utility and elegance are paramount, and in actual practice our current definitions have both, which is exactly why we use them.