Friedman translation

In mathematical logic, the Friedman translation is a certain transformation of intuitionistic formulas. Among other things it can be used to show that the Π⁰₂-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Friedman.

Definition

Let A and B be intuitionistic formulas, where no free variable of B is quantified in A. The translation A^B is defined by replacing each atomic subformula C of A by C ∨ B. For purposes of the translation, ⊥ is considered to be an atomic formula as well, hence it is replaced with ⊥ ∨ B (which is equivalent to B). Note that ¬A is defined as an abbreviation for A → ⊥, hence (¬A)^B = A^B → B.

Application

The Friedman translation can be used to show the closure of many intuitionistic theories under the Markov rule, and to obtain partial conservativity results. The key condition is that the ${\displaystyle \Delta _{0}^{0}}$ sentences of the logic be decidable, allowing the unquantified theorems of the intuitionistic and classical theories to coincide.

For example, if A is provable in Heyting arithmetic (HA), then A^B is also provable in HA.[1] Moreover, if A is a Σ⁰₁-formula, then A^B is in HA equivalent to A ∨ B. This implies that:

• Heyting arithmetic is closed under the primitive recursive Markov rule (MP_PR): if the formula ¬¬A is provable in HA, where A is a Σ⁰₁-formula, then A is also provable in HA.
• Peano arithmetic is Π⁰₂-conservative over Heyting arithmetic: if Peano arithmetic proves a Π⁰₂-formula A, then A is already provable in HA.

See also

• Gödel–Gentzen negative translation

Notes

1. Harvey Friedman. Classically and Intuitionistically Provably Recursive Functions. In Scott, D. S. and Muller, G. H. Editors, Higher Set Theory, Volume 699 of Lecture Notes in Mathematics, Springer Verlag (1978), pp. 21–28. doi:10.1007/BFb0103100