 Bolzano’s proof for the existence of an infinite set is well known among mathematicians, and there are references to it even in Cantor 1883/84, Dedekind 1888, and Russell 1903. Bolzano proves by mathematical induction that there are infinitely many “truths in themselves”, i.e., true propositions. For the induction basis, i.e., that there is at least one true proposition, he uses an indirect proof (as shown already in section 3.3). Bolzano’s induction hypothesis is the assumption that there are at least \(n\) true propositions which we can enumerate as \(S_1, S_2 ,\ldots ,S_n\). Bolzano proves that there must be at least \(n+1\) true propositions by reductio again: If there were not at least \(n+1\) true propositions, i.e., if there were at most \(n\) true propositions, then — according to the induction hypothesis — \(S_1, S_2 ,\ldots ,S_n\) would be the only true propositions; but then there would be another true proposition \(S_{n+1}\), viz. [\(S_1, S_2,\ldots\) and \(S_n\) are the only true propositions], and \(S_1, S_2 ,\ldots ,S_n\) would therefore not be the only true propositions (WL I, 146 f.). Bolzano proposes also a second way for proving this by constructing a series of true propositions in which we add to any true proposition \(S_n\) another true proposition [\(S_n\) is true] that is (due to Bolzano’s criterion for the identity of propositions) different from \(S_n\). Given any true proposition \(S\) whose existence is proved by the induction basis, this will result in an infinite series of true propositions of the following kind: \(S\), [\(S\) is true], [[\(S\) is true] is true] etc. (WL I, 147, Bolzano 1851, § 13). In constructing this series, Bolzano uses a relation \(T\) that can be defined as follows: