 Russell fundamentally conceived of a class as the extension of a concept, or as the extension of a propositional function; indeed, in POM he claims that “a class may be defined as all the terms satisfying some propositional function” (POM, 20). However, Russell was aware already at the time of POM that the supposition there is always a class, understood as an individual entity, as the extension of every propositional function, leads to certain logical paradoxes. Perhaps the most famous, now called “Russell’s paradox”, derived from consideration of the class, w, of all classes not members of themselves. The class w would be a member of itself if it satisfied its defining condition, i.e., if it were not a member of itself. Similarly, w would not be a member of itself if it did not satisfy its defining condition, i.e., if it were a member of itself.  Hence, both the assumption that it is a member of itself, and the assumption that it is not, are impossible. Another related paradox Russell often discussed in this regard has since come to be called “Cantor’s paradox”. Cantor had proven that if a class had n members, that the number of sub-classes that can be taken from that class is 2n, and also that 2n &gt; n, even when n is infinite. It follows from this that the number of subclasses of the class of all individuals, (i.e., the number of different classes of individuals) is greater than the number of individuals.  Russell took this as strong evidence that a class of individuals could not itself be considered an individual. Likewise, the number of subclasses of the class of all classes is greater than the number of members in the class of all classes. This Russell took to be evidence that there is some ambiguity in the notion of a “class” so that the subclasses of the class of “all classes” would not themselves be among its members, as it would seem.