Friedrich Ludwig Gottlob Frege (modern logic, analytic philosophy)
David Hilbert (Hilbert’s program, formalism, ‘’computabilism’’)
Bertrand Russell (type theory of the ‘’principia mathematica’’)
Ernst Zermelo (axiomatic set theory)
Luitzen Egbertus Jan Brouwer (intuitionistic mathematics)
Arend Heyting
Abraham Fraenkel (Zermelo-Fraenkel set theory)
Kurt Friedrich Gödel (completeness theorem, incompleteness theorem)
Alan Turing (proof that the halting problem is not solvable; this recovers Gödel’s incompleteness theorem from computational viewpoint. Turing machine; this is a model for computation)
Willard Van Orman Quine (new foundations; this is a type theory)
Samuel Eilenberg, Saunders Mac Lane (category theory; however it was argued by others that category theory is a foundational theory; see e.g. this)
Francis William Lawvere (category theory, topos theory)
Crispin Wright (neo-Fregean)
Bob Hale (neo-Fregean)
Vladimir Voevodsky (homotopy type theory and univalent foundations)
Harvey Friedman (reverse mathematics)
Rod Nederpelt (weak type theory; related to natural language semantics)
Rod Nederpelt, weak type theory: a formal language for mathematics, pdf
EPSRC project- Theoretical and Implementation advantages of a new lambda notation, (on Automath), web
Freek Wiedijk, the QED manifesto revisited, Nijmegen, 2007, pdf
A foundation for metareasoning part II: the model theory, pdf
homotopy type theory and univalent foundations, web
Steward Shapiro: Foundations without Foundationalism: A Case for Second-order Logic, 2000