Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 1/10/25 2:32 PM, WM wrote:

On 10.01.2025 19:24, joes wrote:

The limit of the ratio is different from the ratio of the limit.

 Let f(x) = 2x/x. The limit of the ratio is 2. The ratio of the limit is undefined. In mathematics we use the limit of the ratio.
 Regards, WM
 

No, we can say that the limit of 2x/x is 2 even at x == 0, where the expression itself is singular. We can then talk about the continous function of the limit of the expression, which is a DIFFERENT function (since it differs at the "value" at x==0) but "close enough" for most things.
Maybe in YOUR naive mathematics you just assume that sort of thing, but when you do, and are dealing with infinite sequences, you get contradictions.
You do know that the sum of an infinite series of terms can change its value based on the order you add up the terms? And thus you need to be careful about how you approach limits of infinite series.