Gödel–Gentzen negative translation
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In proof theory, the Gödel–Gentzen negative translation is a method for embedding classical first-order logic into intuitionistic first-order logic. It is one of a number of double-negation translations that are of importance to the metatheory of intuitionistic logic. It is named after logicians Kurt Gödel and Gerhard Gentzen.
The easiest double-negation translation to describe comes from Glivenko's theorem. It maps each classical formula φ to its double negation ¬¬φ.
Glivenko's theorem states:
- If T is a set of propositional formulas and φ a propositional formula, then T proves φ using classical logic if and only if T proves ¬¬φ using intuitionistic logic.
In particular, a set of propositional formulas is intuitionistically consistent if and only if it is classically satisfiable.
The translation associates with each formula φ in a first-order language another formula φN, which is defined inductively:
- If φ is atomic, then φN is ¬¬φ
- (φ ∧ θ)N is φN ∧ θN
- (φ ∨ θ)N is ¬(¬φN ∧ ¬θN)
- (φ → θ)N is φN → θN
- (¬φ)N is ¬φN
- (∀x φ)N is ∀x φN
- (∃x φ)N is ¬∀x ¬φN
Notice that φN is classically equivalent to φ.
The fundamental soundness theorem states:
- If T is a set of axioms and φ a formula, then T proves φ using classical logic if and only if TN proves φN using intuitionistic logic.
Here TN consists of the double-negation translations of the formulas in T.
Note that φ need not imply its negative translation φN in intuitionistic first-order logic. Troelsta and Van Dalen give a description (due to Leivant) of formulas which do imply their Gödel–Gentzen translation.
There are several alternative definitions of the negative translation. They are all provably equivalent in intuitionistic logic, but may be easier to apply in particular contexts.
One possibility is to change the clauses for disjunction and existential quantifier to
- (φ ∨ θ)N is ¬¬(φN ∨ θN)
- (∃x φ)N is ¬¬∃x φN
Then the translation can be succinctly described as: prefix ¬¬ to every atomic formula, disjunction, and existential quantifier.
Another possibility (described by Kuroda) is to construct φN from φ by putting ¬¬ before the whole formula and after every universal quantifier. Notice that this reduces to the simple ¬¬φ translation if φ is propositional.
It is also possible to define φN by prefixing ¬¬ before every subformula of φ.
The double-negation translation was used by Gödel (1933) to study the relationship between classical and intutionistic theories of the natural numbers ("arithmetic"). He obtains the following result:
- If a formula φ is provable from the axioms of Peano arithmetic then φN is provable from the axioms of intuitionistic Heyting arithmetic.
This result shows that if Heyting arithmetic is consistent then so is Peano arithmetic. This is because a contradictory formula θ ∧ ¬θ is interpreted as θN ∧ ¬θN, which is still contradictory. Moreover, the proof of the relationship is entirely constructive, giving a way to transform a proof of θ ∧ ¬θ in Peano arithmetic into a proof of θN ∧ ¬θN in Heyting arithmetic. (By combining the double-negation translation with the Friedman translation, it is in fact possible to prove that Peano arithmetic is Π02-conservative over Heyting arithmetic.)
The propositional mapping of φ to ¬¬φ does not extend to a sound translation of first-order logic, because ∀x ¬¬φ(x) → ¬¬∀x φ(x) is not a theorem of intuitionistic predicate logic. This explains why φN has to be defined in a more complicated way in the first-order case.
- J. Avigad and S. Feferman (1998), "Gödel's Functional ("Dialectica") Interpretation", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9
- S. Buss (1998), "Introduction to Proof Theory", Handbook of Proof Theory, S. Buss, ed., Elsevier. ISBN 0-444-89840-9
- G. Gentzen (1936), "Die Widerspruchfreiheit der reinen Zahlentheorie", Mathematische Annalen, v. 112, pp. 493–565 (German). Reprinted in English translation as "The consistency of arithmetic" in The collected papers of Gerhard Gentzen, M. E. Szabo, ed.
- K. Gödel (1933), "Zur intuitionistischen Arithmetik und Zahlentheorie", Ergebnisse eines mathematische Kolloquiums, v. 4, pp. 34–38 (German). Reprinted in English translation as "On intuitionistic arithmetic and number theory" in The Undecidable, M. Davis, ed., pp. 75–81.
- A. N. Kolmogorov (1925), "O principe teritium non datur" (Russian). Reprinted in English translation as "On the principle of the excluded middle" in From Frege to Gödel, van Heijenoort, ed., pp. 414–447.
- A. S. Troelsta (1977), "Aspects of Constructive Mathematics", Handbook of Mathematical Logic", J. Barwise, ed., North-Holland. ISBN 0-7204-2285-X
- A.S. Troelsta and D. van Dalen (1988), Constructivism in Mathematics. An Introduction, volumes 121, 123 of Studies in Logic and the Foundations of Mathematics, North–Holland.
- "Intuitionistic logic", Stanford Encyclopedia of Philosophy.