Short answer: Gödel was “pure” in a metaphysical sense (strong Platonism) but not a purist about restrictive proof methods. He embraced full classical logic, non-constructive existence proofs, and strong set-theoretic principles when they clarified truth—so, philosophically pure, methodologically liberal.
Here are the main senses of “purity,” and where Gödel—and others—fit:
Constructive/predicative purists (restrict logic):
L. E. J. Brouwer (intuitionism, rejects LEM), Errett Bishop (constructive analysis), early Hermann Weyl (Das Kontinuum, predicativism), Poincaré (anti-impredicativity), Kronecker (arithmetization).
Gödel: Not a purist here—he defended classical logic and accepted non-constructive methods. His incompleteness theorems and the constructible universe L (consistency of CH and AC) are classical, not constructive.
Formalist/finitist purists (only finitary consistency proofs):
Hilbert (program), Bernays, Herbrand, Gentzen.
Gödel: Again not a purist—his results undermined Hilbert’s strict program.
Proof-theoretic “purity of method” (eliminate detours):
Gentzen (cut-elimination), Prawitz (normalization). This is a technical “purity” ideal about analytic proofs, not moral or metaphysical.
Gödel: Strong yes. He is the textbook mathematical Platonist—convinced of objective mathematical reality and the meaningfulness of set-theoretic questions like CH. He favored adding new axioms if they reflect that reality.
Others aligned with this kind of purity:
Cantor (infinite hierarchies, transfinite numbers), Zermelo (axiomatizing set theory to secure pure reasoning), many modern set theorists (Solovay, Woodin) who pursue large cardinals and truth-seeking axioms. (Methods vary—Cohen’s forcing is methodologically liberal but often in service of the same truth-seeking ideal.)