Abstract
The foundation scheme in set theory asserts that every nonempty class has an $$\in $$ź-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and $$\alpha $$ź-recursion theory. We take KP set theory without foundation (called KP$$^-$$-) as the base theory. We show that KP$$^-$$- + $$\Pi _1$$ź1-Foundation + $$V=L$$V=L is enough to carry out finite injury arguments in $$\alpha $$ź-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.
Highlights
Denote by KP− the theory obtained from the usual Kripke–Platek set theory KP
An investigation of the logical strength of fragments of KP can be found in Ressayre’s notes [21], where he showed that the hierarchy of n-foundation schemes is strict
Rathjen gave [19] a proof-theoretic analysis of primitive recursive set functions in the axiom system of KP− + infinity + 1-foundation, and characterized the logical strength of KP− + infinity + n+2-foundation by the smallest ordinal α such that Lα is a model of all 2 sentences provable in the theory [18]
Summary
Kripke–Platek set theory (KP) consists of the Extensionality, Foundation, Pairing and Union axioms together with 0-Separation and 0-Collection:. (ii) Foundation: If y is not a free variable in φ(x), [∃xφ(x) → ∃x(φ(x) ∧ ∀y ∈. (iii) Pairing: ∀x, y∃z(x ∈ z ∧ y ∈ z). (v) 0-Separation: ∀x∃y∀z(z ∈ y ↔ (z ∈ x ∧ φ(z))) for each 0 formula φ. KP does not contain the Infinity axiom. For every class of formulas, -Induction holds if and only if ¬ -Foundation holds, where ¬ = {¬φ : φ ∈ }. We use KP− to denote KP without Foundation (i.e. Clauses (i), (iii)–(vi)). (1) Strong Pairing: ∀x, y∃z (z = {x, y}). (4) Strong 1-Collection: Suppose f is a 1 function. (5) Ordered Pair: ∀x, y∃z (z = (x, y)).
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