1, 2, 3, 4, 5, 6, 7, 8, 1, 1
As we see, N is not equal to the nth number of E, and as n is arbitrary, N does not appertain to E and yet N should belong to this assemblage since we have defined it with a finite number of words.
We shall later see that M. Richard has himself given with much sagacity the explanation of his paradox and that this extends, mutatis mutandis, to the other like paradoxes. Again, Russell cites another quite amusing paradox: What is the least whole number which can not be defined by a phrase composed of less than a hundred English words?
This number exists; and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the English language are not infinite in number. Therefore among them will be one less than all the others. And, on the other hand, this number does not exist, because its definition implies contradiction. This number, in fact, is defined by the phrase in italics which is composed of less than a hundred English words; and by definition this number should not be capable of definition by a like phrase.
VI
Zigzag Theory and No-class Theory
What is Mr. Russell's attitude in presence of these contradictions? After having analyzed those of which we have just spoken, and cited still others, after having given them a form recalling Epimenides, he does not hesitate to conclude: "A propositional function of one variable does not always determine a class." A propositional function (that is to say a definition) does not always determine a class. A 'propositional function' or 'norm' may be 'non-predicative.' And this does not mean that these non-predicative propositions determine an empty class, a null class; this does not mean that there is no value of x satisfying the definition and capable of being one of the elements of the class. The elements exist, but they have no right to unite in a syndicate to form a class.
But this is only the beginning and it is needful to know how to recognize whether a definition is or is not predicative. To solve this problem Russell hesitates between three theories which he calls
A. The zigzag theory;
B. The theory of limitation of size;
C. The no-class theory.
According to the zigzag theory "definitions (propositional functions) determine a class when they are very simple and cease to do so only when they are complicated and obscure." Who, now, is to decide whether a definition may be regarded as simple enough to be acceptable? To this question there is no answer, if it be not the loyal avowal of a complete inability: "The rules which enable us to recognize whether these definitions are predicative would be extremely complicated and can not commend themselves by any plausible reason. This is a fault which might be remedied by greater ingenuity or by using distinctions not yet pointed out. But hitherto in seeking these rules, I have not been able to find any other directing principle than the absence of contradiction."