In mathematical logic, the Friedman translation is a certain transformation of formulas. Among other things it can be used to show that the ÃÂ <sup>0</sup><sub style="margin-left:-0.65em">2</sub>-theorems of various first-order theories of classical mathematics are also theorems of intuitionistic mathematics. It is named after its discoverer, Harvey Friedman.
Let A and B be formulas, where no free variable of B is quantified in A. The translation A<sup>B</sup> is defined by replacing each atomic subformula C of A by . For purposes of the translation, âÂÂ¥ is considered to be an atomic formula as well; hence it is replaced with (which is equivalent to B). The translation is relevant in an intuitionistic context and here ìA is generally defined as an abbreviation for hence
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 -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<sup>B</sup> is also provable in HA. Moreover, if A is a ã<sup>0</sup><sub style="margin-left:-0.65em">1</sub>-formula, then A<sup>B</sup> is in HA equivalent to . By setting B = A, this implies that: