We were taught about countably infinite and uncountably infinite sets in my first year at university. It's hard to wrap your head around at first, but it does make sense. The way we were taught was:
The set of all natural numbers (i.e. all the positive whole numbers from 0 to infinity) are countable, since you can theoretically count them one by one forever. There is a specific start value (0) and the next number in the sequence is well defined (1).
The set of real numbers (i.e. any non-imaginary / non-complex number) are not countably infinite, since there is no specific start value (negative infinity is not a number, but a limit), and even if we were to define a start number (say, 0), the number that comes after 0 is not defined. In fact, for any number you can come up with, there exists a number which fits between it and 0. For instance, if we were to pick 1 as the next number, we find that 0.5 fits between them, so 1 cannot be the next number in our count. If we use 0.5, we find that 0.25 fits between them...and so on.
Using standard notation to try and represent the first number that comes after 0 gets us into even more trouble, since 0.000....1 is an invalid number (you can't have an infinite amount of 0's followed by a 1). This renders the entire set of real numbers uncountable, and ultimately leads to all the lovely "impossible" examples of mathematics you posted about.
The set of all natural numbers (i.e. all the positive whole numbers from 0 to infinity) are countable, since you can theoretically count them one by one forever. There is a specific start value (0) and the next number in the sequence is well defined (1).
The set of real numbers (i.e. any non-imaginary / non-complex number) are not countably infinite, since there is no specific start value (negative infinity is not a number, but a limit), and even if we were to define a start number (say, 0), the number that comes after 0 is not defined. In fact, for any number you can come up with, there exists a number which fits between it and 0. For instance, if we were to pick 1 as the next number, we find that 0.5 fits between them, so 1 cannot be the next number in our count. If we use 0.5, we find that 0.25 fits between them...and so on.
Using standard notation to try and represent the first number that comes after 0 gets us into even more trouble, since 0.000....1 is an invalid number (you can't have an infinite amount of 0's followed by a 1). This renders the entire set of real numbers uncountable, and ultimately leads to all the lovely "impossible" examples of mathematics you posted about.
