Proof-theoretical analysis: weak systems of functions and classes

Proof-theoretic conservations of weak weak intuitionistic constructive set theories

“…Furthermore [11] posed a deeper problem conjecturing that intuitionistic set theories under consideration are conservative extensions of the underlying arithmetical intuitionistic formalisms. These conjectures (et al) have been confirmed [14] for Friedman's extensional set theories T 1 , T 2 , T 3 having proof-theoretic strengths |T 1 | = ε 0 , |T 2 | = ϕ ε 0 (0), |T 3 | = ϕ ε Ω+1 (0) (also known as Howard ordinal |ID 1 |). Moreover [14] strengthened Friedman's conjectures by also proving conservations in the presence of consistent combinations of other constructive principles like an anti-foundation axiom Cpl and/or finite-types axiom of choice AC ft [14,Corollary 3,p.…”

“…Bisimulation + realizability translation [13,14] of a given set theory T (here Basic (i) + Ext + Δ 0 -Sep + Θ (+SC) + (either Fnd or SClps)) into appropriate Feferman-style applicative intensional intuitionistic theory of functions and classes (call it T app ). 2.…”

“…35]. Actually for every weak set theory S in question [14] expressed the solution in the "most conservative" form S A ⇔ HA + TI (< |S|) A, for any arithmetical statement A, where TI (< |S|) denotes the arithmetical transfinite induction scheme below proof theoretic ordinal of S. The proofs were based on the author's constructive semantics [13] of analogous weak set theories.…”

“…The proof techniques are the same as those used in [13,14]. In the sequel we adopt basic notations and abbreviations used in [19] and often use superscript (i) to emphasize that underlying formal system is based on the intuitionistic logic.…”

Arithmetical conservation results

“…Beeson showed how this can be understood as the composition of forcing and realizability and extended Goodman's theorem to the extensional setting (see [2] and also [3]). Other proofs have been given by Gordeev [9], Mints [18], Coquand [4] and Renardel de Lavalette [21]; the authors of this paper are unsure whether this list is complete. (An interesting observation, due to Kohlenbach, is that Goodman's Theorem can fail badly for fragments: see [12].…”