The Diagonalization Lemma (= the fixed point lemma) requires something like the Robinson Arithmetic Q. The diagonalization function must be (strongly) representable. In the language of set theory, the very weak "baby set theory" i.e. "the adjunctive set theory" will do.

"Descriptive set theory" is a field of mathematics related to topology and was initiated by the French semi-intuitionists (Lebesgue, Baire, Borel). 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. It is common to count it, in the literature on the foundations of mathematics, the name notwithstanding, as "ordinary mathematics", in contradistinction to set theory.

I beg to differ. There are true $\Sigma_2^0$ sentences which are not provable even in $T_1$.