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The next type consists of classes or ranges of individuals; then one has classes of classes of objects of the lowest type, and so on (see also the entry on type theory)..
New difficulties still arise if one accepts that propositions form a type (as they are the only objects of which it can be meaningfully asserted that they are true or false).
Let \(m\) be a class of propositions and let \(\Pi m\) be the proposition “every proposition of \(m\) is true” (regarded as a possibly infinitary conjunction); then, if \(m\) and \(n\) are different classes, the propositions \(\Pi m\) and \(\Pi n\) are different, i.e., the map associating to \(m\) its product \(\Pi m\) is injective.
Therefore, if we consider the class \[ \ = R, \] we have, by injectivity, a contradiction.
Russell was at first entangled in the study of “the contradictions in the relation of continuous quantity to number and the continuum” (cited in Moore 1995, p. The effect of the antinomy is that it is impossible to have an abstraction operation \(\phi \mapsto \) mapping injectively any concept (property) \(\phi\) into its extension (the class of all \(x\) such that \(\phi(x))\) (i.e., so that if the classes defined by \(\phi\) and \(\psi\) are equal, then \(\phi(a)\leftrightarrow \psi(a)\), for every object \(a)\).
The father of set theory, Cantor, had noticed similar difficulties already in 1895 (as witnessed by Bernstein and by letters to Hilbert and Dedekind).
Indeed, in a second letter to Dedekind of August 31, 1899 Cantor pointed out another problem, involving the notion of the cardinal number and implying that one cannot consistently think of the “the set of all conceivable sets”, say \(M\).
Several basic notions of logic, as it is presently taught, have reached their present shape at the end of a process which has been often triggered by various attempts to solve paradoxes.
This is especially true for the notions of (logical languages of a given order, the notion of satisfiability, definability).
In the case of the difficulty discovered by Burali-Forti, the consequence for Cantor was that the multiplicity () of ordinal numbers is itself well-ordered, but is not a set: hence, no ordinal can be assigned to it, and the antinomy is resolved.