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The rational numbers can be put onto a list that just runs down the page. Therefor they are only one dimensionally infinite. The irrationals require at least two dimensions of infinite listing.
For any who don't get what we are joking about here:
Take any rational number in the form of:
ABCD..../UVWX....
Prepend on either number enough zeros to pad the number of digits to be equal.
Prepend a "1" on each number.
Interleave the digits of this number like this:
AUBVCW....
You now have an integer value that will be different for every rational you started with.
This integer is the index for where to write down this rational number on the list.
Because of the prepending of "1", we know that the first two digits of this index will always be "11". This means that we will have room left over for 98 times as many more numbers.
****************************Assume that you have a list of the irrationals.
Assume that they are padded on the right with many zeros below the decimal or consider white spaces as zeros.
From the first number, take the first digit to the right of the decimal. From the second take the second. Continue down the list creating a string of digits.
For each of these selected digits add one with no carry. 0 becomes 1,
1 becomes 2 .... 9 becomes 0Place a decimal point in front of this string of digits.
You now have a number between 0 and 1 that differs from the first in the first digit and differs from the second on the second digit and so on. Therefor this number must not be on our proposed list. This proves that in a list that is infinite in only one direction the irrationals can't be listed.
It is left to the reader to prove whether or not there exists a rule that would allow all the irrationals to be placed in two dimensions.