_The True Solution_
What choice ought we to make among these different theories? It seems to me that the solution is contained in a letter of M. Richard of which I have spoken above, to be found in the _Revue générale des sciences_ of June 30, 1905. After having set forth the antinomy we have called Richard's antinomy, he gives its explanation. Recall what has already been said of this antinomy. _E_ is the aggregate of _all_ the numbers definable by a finite number of words, _without introducing the notion of the aggregate E itself_. Else the definition of _E_ would contain a vicious circle; we must not define _E_ by the aggregate _E_ itself.
Now we have defined _N_ with a finite number of words, it is true, but with the aid of the notion of the aggregate _E_. And this is why _N_ is not part of _E_. In the example selected by M. Richard, the conclusion presents itself with complete evidence and the evidence will appear still stronger on consulting the text of the letter itself. But the same explanation holds good for the other antinomies, as is easily verified. Thus _the definitions which should be regarded as not predicative are those which contain a vicious circle_. And the preceding examples sufficiently show what I mean by that. Is it this which Russell calls the 'zigzaginess'? I put the question without answering it.
