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Show 8 more comments. Active Oldest Votes. Improve this answer. Add a comment. Therefore, in order to counter concerns raised by the discovery of the logical and set-theoretic paradoxes, a new approach was needed to justify modern mathematical methods. By the summer of , Hilbert had formulated such an approach. First, modern mathematical methods were to be represented in formal deductive systems. The approach became known as Hilbert's program.
The epsilon calculus was to provide the first component of this program, while his epsilon substitution method was to provide the second.
The extension is conservative in the sense that it does not add any new first-order consequences. But, conversely, quantifiers can be defined in terms of the epsilons, so first-order logic can be understood in terms of quantifier-free reasoning involving the epsilon operation.
It is this latter feature that makes the calculus convenient for the purpose of proving consistency. Suitable extensions of the epsilon calculus make it possible to embed stronger, quantificational theories of numbers and sets in quantifier-free calculi. Hilbert expected that it would be possible to demonstrate the consistency of such extensions.
In his Hamburg lecture in , Hilbert first presented the idea of using such an operation to deal with the principle of the excluded middle in a formal system for arithmetic. This section will describe a version of the calculus corresponding to first-order logic, while extensions to first- and second-order arithmetic will be described below.
The only rules of the calculus are the following:. Note that the calculus just described is quantifier-free. The converse is, however, not true: not every formula in the epsilon calculus is the image of an ordinary quantified formula under this embedding. Hence, the epsilon calculus is more expressive than the predicate calculus, simply because epsilon terms can be combined in more complex ways than quantifiers.
It is worth noting that epsilon terms are nondeterministic. For many applications, however, this additional schema is not necessary. This includes a discussion of the first and second epsilon theorems with applications to first-order logic, the epsilon substitution method for arithmetic with open induction, and a development of analysis that is, second-order arithmetic with the epsilon calculus. Since the epsilon calculus includes first-order logic, the first epsilon theorem implies that any detour through first-order predicate logic used to derive a quantifier-free theorem from quantifier-free axioms can ultimately be avoided.
The second epsilon theorem shows that any detour through the epsilon calculus used to derive a theorem in the language of the predicate calculus from axioms in the language of the predicate calculus can also be avoided. More generally, the first epsilon theorem establishes that quantifiers and epsilons can always be eliminated from a proof of a quantifier-free formula from other quantifier-free formulae.
Hilbert and Bernays use this theorem to give a finitary consistency proof of elementary geometry , Sec 1. The difficulty for giving consistency proofs for arithmetic and analysis consists in extending this result to cases where the axioms also contain ideal elements, i.
Further reading. Leisenring is a relatively modern book-length introduction to the epsilon calculus in English. The first and second epsilon theorem are described in detail in Zach The original proofs are given for axiomatic presentations of the epsilon-calculus. Maehara was the first to consider sequent calculus with epsilon terms. He showed how to prove the second epsilon theorem using cut elimination, and then strengthened the theorem to include the schema of extensionality Maehara Baaz et al.
Hilbert and Bernays used the methods of the epsilon calculus to establish theorems about first order logic that make no reference to the epsilon calculus itself. To understand the two parts of the theorem below, it helps to consider a particular example. Moreover, and this is seldom recognized, whereas the proof based on cut-elimination provides a bound on the length of the Herbrand disjunction only as a function of the cut rank and complexity of the cut formulas in the proof, the length obtained from the proof based on the epsilon calculus provides a bound as a function of the number of applications of the transfinite axiom, and the rank and degree of the epsilon-terms occurring therein.
In other words, the length of the Herbrand disjunction depends only on the quantificational complexity of the substitutions involved, and, e. It is here that epsilon methods play a central role. 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 Sieg A model-theoretic version of this is discussed in Avigad a.
As noted above, historically, the primary interest in the epsilon calculus was as a means to obtaining consistency proofs. Similarly, one can prove the consistency of predicate logic or the pure epsilon calculus , by specializing to interpretations where the universe of discourse has a single element.
These considerations suggest the following more general program for proving consistency:. For that, it suffices to show that, given any finite set of closed instances of axioms, one can assign numerical values to terms in such a way that all the axioms are true under the interpretation. If it does, all critical formulas are true formulas without epsilon-terms. The challenge was to extend the approach to more than one epsilon term, to nested epsilon terms, and ultimately to second-order epsilons in order to obtain a consistency proof not just of arithmetic, but of analysis.
The difficulty in dealing with nested epsilon terms can be described as follows. Show 2 more comments. Active Oldest Votes. Chris Taylor Chris Taylor 27k 4 4 gold badges 74 74 silver badges bronze badges. The more you know This is why often you hear people talk about "epsilon relation" or "epsilon induction".
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