While reading Paul Erdos' story in "My brain is open" I found Cantor's diagonal argument which he used to 'prove' that the decimals between 0 and 1 are uncountable. What's even worse, he used that to say that the infinity of the decimals is larger than the infinity of the integers (but that's another matter). What I would like to know is how could he establish the diagonal argument on a list that I'm not sure can be created at all. How can someone create a list like:
1 <--> .2332245..... 2 <--> .4898495.....
The basic question here is how can you assign the number 1 to a number that has an infinite number of digits. In the example above, what is the number that 1 'points' to? Since you can keep adding an infinite number of digits after the decimal point how can you say that 1 is pointing to a specific number? A possible list would be something like:
1 <--> .25 2 <--> .50
Here the number 1 is assigned to the specific number .25, not to an arbitrary number that has an infinite number of digits after the decimal point.
Given these questions, how can the proof itself hold true. I don't see that its initial premise makes logical sense. What's also confusing is saying that if you can't use the integers to count the decimals that kind of assumes that you'll run out of integers (you'll reach the largest integer ever) which makes no sense. Comments please!!
In an alternate but related topic, I think it is absurd to talk about an infinite being larger than another. That just does not make sense at all. Infinity doesn't have an end so:
