 There are some concrete examples of mathematical statements undecided even in stronger theories which come from the so-called descriptive set theory. This field of mathematics is related to topology and was initiated by the French semi-intuitionists (Lebesgue, Baire, Borel; see the section on descriptive set theory, etc., in the entry on  intuitionism in the philosophy of mathematics).  It studies sets which possess relatively simple definitions (in contradistinction to the ideas of arbitrary sets and various higher power-sets, which the semi-intuitionists rejected as meaningless) called projective or analytic sets. Classically these were defined as the sets that can be built up from a countable intersection of open sets by taking continuous images and complements finitely many times; they coincide with the sets which are definable in the language of P\(^2\). In particular, the so-called Borel sets can be simply defined both by a formula of the form \(\exists XA(x)\) and by a formula of the form \(\forall XB(x)\), where \(A\) and \(B\) do not contain any set variables (in logician’s terminology, Borel sets are the \(\Delta^{1}_1\) sets). A Borel function is defined analogously (see, e.g., Martin 1977).