P I   C C  P  M

Hilbert’s original aim, to show the consistency of the use of such ideal elements by finitistic means, which would suffice to eliminate these ideal elements from proofs of purely universal (‘real’) theorems (‘Hilbert’s program’), turned out to be impossible for theories containing a sufficient amount of number theory by K. Gödel’s 2nd incompleteness theorem. Nevetheless, various partial realizations of this program and many relative consistency proofs could be achieved. One class of tools used to obtain this are so-called proof interpretations I which transform proofs p in theories T of theorems A into new proofs pI in theories T I of the interpretation AI of A. If (0 = 1)I ≡ (0 = 1), then this yields a consistency proof for T relative to the assumed consistency of T I . Whereas, Hilbert’s program focusses exclusively on (ideal proofs of) purely universal theorems, a natural ‘shift of emphasis’ (G. Kreisel) is to try to apply proof interpretations to interesting proofs of existential theorems with the goal to extract new information from the proof which was not visible beforehand.1 This new information often consists of effective data such as algorithms or effective bounds (extracted from prima facie ineffective proofs) but also continuous dependence or even full independence of