This isn't precisely a paradox. But the result comes from deriving a contradiction.

This is an attempt to produce a list of the real numbers which are claimed to be unlistable. I will follow the standard proof with an interesting counterexample. It suffices to limit our scope to the numbers between zero and one.

First: List the real numbers in their decimal expansions.
Second: Beginning with the first digit of the first number, if it is equal to zero, write on another paper the number one after a decimal point. Otherwise write a zero.
Third: Go to the second digit of the second number. Again, if it equals zero, then write the number one on the other sheet next to the number we've begun to build. Otherwise write zero.
Fourth: Go to the third number and so on.

The conclusion: The real numbers are NOT listable because we posited a list with all the real numbers on it and then constructed a number not on the list.

Counter-example: Consider the following list construction: next to decimal points, list the numbers zero through nine in order. In other words, .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Next to each of these, write the digits zero through nine again. In other words, .00 .01 ... .09 .10 .11 ... .19 ... .99 and so forth. This list seems to create all real numbers. The salient argument lied in not considering the above as a list. But if you can't construct my list with an "infinite" process, then you can't construct the Cantor number in the diagonalization proof.

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