Linked bibliography for the SEP article "The Epsilon Calculus" by Jeremy Avigad and Richard Zach

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The following list of references provides a starting point for exploring the literature, but is by no means comprehensive.

Hilbert's Program

The following source books have many of the original papers:

Overviews of the historical development of logic and proof theory in the Hilbert school can be found in

The Early History of the Epsilon Calculus and Epsilon Substitution Method

The original work:

Ackermann's 1940 proof is discussed in

Maehara showed how to prove the second epsilon theorem using cut elimination, and then strengthened the theorem to include the schema of extensionality, in

An early application of epsilon substitution is Georg Kreisel's no-counterexample interpretation.

The following provide modern presentations of Hilbert's epsilon calculus, not just from an introductory standpoint:

Corrections to errors in the literature (including Leisenring's book) can be found in

A variation of the epsilon calculus based on Skolem functions, and therefore compatible with first-order logic, is discussed in

General References for Proof Theory

The following contains a number of proof-theoretic results that are proved using methods similar to the ones used by Hilbert, Bernays, and Ackermann, though using Skolem functions instead of epsilon terms:

For more on ordinal analysis, see, for example:

Herbrand's Theorem

Herbrand's theorem originally appeared in

English translations can be found in van Heijenoort (see above), and

Further historical information can be found in

The literature on Herbrand's theorem is vast. For some general overviews, in addition to the general proof-theoretic references above, see

A striking application of Herbrand's theorem and related methods is found in Luckhardt's analysis of Roth's theorem:

For a discussion of useful extensions of Herbrand's methods, see

A model-theoretic version of this is discussed in

More Recent Developments in the Epsilon Substitution Method

In the following two papers, William Tait analyzed the epsilon substitution method in terms of continuity considerations:

More streamlined and modern versions of this approach can be found in:

The following paper shows that the epsilon substitution method for first-order arithmetic is, in fact, strongly normalizing:

Connections between cut elimination and epsilon substitution method are explored in

The epsilon substitution method has been extended to predicative fragments of second-order arithmetic in:

The following papers address impredicative theories:

A development of set theory based on the epsilon-calculus is given by

Epsilon Operators in Linguistics, Philosophy, and Non-classical Logics

The following is a list of some publications in the area of language and linguistics of relevance to the epsilon calculus and its applications. The reader is directed in particular to the collections von Heusinger and Egli (2000) and von Heusinger et al. (2002) for further discussion and references.

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