Grue 2001 presents a very strong extension of lambda calculus, the so-called map theory, in which standard axiomatic set theory becomes interpretable, and which can be used to shed light on the difference between Russell's and Burali-Forti's paradoxes.
The hope for metamathematics was that, by using only intuitively convincing, weak number-theoretic arguments (called finitary methods), unimpeachable proofs of the consistency of such theories as axiomatic set theory could be given.
It takes a bit of work to use the simpler standard formulation to derive the Recursion Principle, and this is one of the basic topics in any course on axiomatic set theory.
…set theory, and, in fact, axiomatic set theory is often regarded as a part of logic.
Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations.
The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-Gödel.
This work appeared in a series of articles under the title "A System of Axiomatic Set Theory" (1937 54), from which the principal theses were published as Axiomatic Set Theory (1958).
The foundations of axiomatic set theory are in a state of significant change as a result of new discoveries.
Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory.
It is usually said that all of mathematics can, in principle, be formulated in a sufficiently theorem-rich system of axiomatic set theory.
