Comparative Historical Studies (3): Wittgenstein’s Anticipation of Church Numerals

Wittgenstein’s treatment of natural numbers as exponents of an operation in his Tractatus clearly anticipates Church numerals as it has been showed by P. Frascolla in his careful reconstruction of Tractatus 6.02 (and other passages). Another contribution by Wittgenstein (in another work) is his question about “ \(f(f)\) ” (which is answered in the \(\lambda \) -calculus by the self-application function \(\lambda f\) .( \(f f))\) . The starting point of Wittgenstein’s fundamental thesis according to which a number is the exponent of an operation (in Tractatus 6.021) is his inductive definition of the operation variable \(\Omega \) . The variable x in \(\Omega ^{0}x\) used by Wittgenstein corresponds to the generic argument a in Church numerals (like a in the Church numeral \(\lambda f\) . \(\lambda a\) .( \(f \quad a))\) and the successive application of the operation \(\Omega \) in \(\Omega \Omega x\) , \(\Omega \Omega \Omega x\) , etc. corresponds to the successive application of the generic function f in Church numerals (like f in \(\lambda f\) . \(\lambda a\) . \(f(f \quad a)\) , \(\lambda f\) . \(\lambda a\) . \(f(f(f \quad a))\) , etc.). In this way, \(\Omega ^{SS...S0}x\) corresponds to the n-th Church numeral \(n \) for \(n \ge \) 0 occurrences of “S” (the successor function). Then, according to this reconstruction, the language of the general theory of logical operations in the Tractatus is correlated with the language of arithmetic \(\lambda \) -terms.