"Some real numbers are not definable, by Cantor's diagonal argument."

There are subtleties involved in formalizing the statement "some real numbers are not definable", as explained in [Joel's answer][1] to [this question][2].  The statement can be seen to hold in some models and fail in other models.  However, the claim that the statement *follows from Cantor's diagonal argument* is clearly false, yet seems to be fairly common.

The false reasoning typically proceeds in three steps:

1. There are only countably many definitions of real numbers: $\varphi_0(x),\varphi_1(x),\ldots$ (this part is ok.)

2. Consider the countably many real numbers so defined: $x_0,x_1,\ldots$ (this part is problematic for subtle reasons.)

3. Use Cantor's diagonal argument to obtain a real number $y$ that is not in the sequence from step 2, and is therefore not definable.

For the moment, let us assume that step 2 succeeds in the way that one might naively think it would. Then we have *defined* a sequence $x_0,x_1,\ldots$ containing all definable real numbers.  Therefore Cantor's diagonal argument in step 3 *defines*, from this sequence, a real number $y$ that is not in the sequence. So $y$ is both definable and not definable, and we obtain a contradiction outright!  Clearly, something is wrong (and it turns out to be in step 2.)

  [1]: https://mathoverflow.net/a/44129/1682
  [2]: https://mathoverflow.net/q/44102/1682