Intuitionistic fixed point theories over set theories

Abstract

In this paper we show that the intuitionistic fixed point theory FiX i (T ) over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.1 Intuitionistic fixed point theory over set theories T For a theory T in a laguage L, let Q(X, x) be an X-positive formula in the language L ∪ {X} with an extra unary predicate symbol X. Introduce a fresh unary predicate symbol Q together with the axiom stating that Q is a fixed point of Q(X, x):By the completeness theorem, it is obvious that the resulting extension of T is conservative over T , though it has a non-elementary speed-up over T when T is a recursive theory containing the elementary recursive arithmetic EA, cf.[3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q. Then FiX(PA) is stronger than PA, e.g., FiX(PA) proves the consistency of PA. For the proof-theretic strength of the fixed point theory FiX(PA), see [2,10,16].On the other side, W. Buchholz [15] shows that an intuitionistic fixed point theory over the intuitionistic (Heyting) arithmetic HA for strongly positive formulae Q(X, x) is proof-theoretically reducible to HA. In a language of arithmetic strongly positive formulae with respect to X are generated from arithmetic formulae and atomic ones X(t) by means of positive connectives ∨, ∧, ∃, ∀. Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope 1

Rumfitt on truth-grounds, negation, and vagueness

“…The second is this: If the ancestral operator is strong enough to make quantification over the natural numbers determinate in the relevant sense, it would be expected that the mere addition of the ancestral operator to Heyting arithmetic would already result in a stronger system, as it does in the classical case. However, adding the ancestral to Heyting arithmetic is conservative (Arai 2010).…”

Lifting proof theory to the countable ordinals II: second-order indescribable cardinals

“…An intuitionistic fixed point theory FiX i (ZFLK k,n ) over ZFLK k,n is introduced in [6], and shown to be a conservative extension of ZFLK k,n .…”

Cut-elimination for $ω_{1}$

“…Hence the whole proof is formalized in an intuitionistic fixed point theory Fix i (ZF) over ZF, which is a conservative extension of ZF, cf. [4]. Theorem 1.1 aims at enlarging the realm of the ordinal analysis, a topic in proof theory.…”