 It is often stated that before the celebrated Paris-Harrington theorem (see below), no such natural independent mathematical statements were known. This is not, however, strictly speaking, correct. Already much earlier, around 1935, Gerhard Gentzen (see the entry on the  development of proof theory)  had provided such a statement. It is very natural to generalize the idea of induction from the domain of natural numbers to the domain of ordinal numbers. In set theory, such generalizations are called principles of transfinite induction. Though some constructivists may be sceptical about the legitimacy of full set theory, there are limited and more concrete cases of transfinite induction (only dealing with some well-defined classes of countable ordinals) that are perfectly acceptable even from the constructivist or intuitionist viewpoint. One important case is the principle of transfinite induction up to the ordinal called \(\varepsilon_0\). Gentzen showed that the consistency of PA can be proved if this transfinite induction principle is assumed. Therefore, because of the second incompleteness theorem, the principle itself cannot be provable in PA (Gentzen 1936).