A sharpened version of McAloon's theorem on initial segments of models of IΔ0

D'Aquino, Paola

doi:10.1016/0168-0072(93)90197-l

Cited by 8 publications

(13 citation statements)

References 4 publications

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“…In this section we give a generalization of Ackermann's function. Similar constructions have been considered by P. D'Aquino (see [1]), R. Kaye (see [9]) and R. Sommer (see [18]). The aim of the definition we develop here is to describe inductive n -functional subtheories of I n+1 .…”

Section: Ackermann's Functionssupporting

confidence: 52%

“…In section 5 (see also [1] or [18]) it will be proved that there exists ϕ(u, x, y) ∈ 0 such that Let Ack = {ϕ(k, x, y) : k ∈ ω}. It holds that: By 1.7.3-(i), if we add to I 0 axioms expressing that each primitive recursive function is total, then we obtain induction for every 1 (I 1 )-formula.…”

Section: ((3) ⇒ (1))mentioning

confidence: 99%

“…For each level of Grzegorczyk's hierarchy, E k , k ≥ 3, (see [17]) we have a similar result using the theory I 0 + ∀x ∃y [F 0,k−2 (x) = y] (see 4.6). So, if I 0 is extended with axioms asserting that each function in E k is total, then we obtain induction for every 1…”

Section: ((3) ⇒ (1))mentioning

confidence: 99%

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On the quantifier complexity of ? n+1 (T)? induction

Cordón‐Franco¹,

Margarit²,

Lara‐Martín

2004

Archive for Mathematical Logic

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In this paper we continue the study of the theories I n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I n+1 (T) is n+2-axiomatizable. In particular, I n+1 (I n+1) gives an axiomatization of Th n 2 (I n+1) and is not finitely axiomatizable. This fact relates the fragment I n+1 (I n+1) + to induction rule for n +1-formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons' conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for n+2 and n+1 formulas (see [2]). In [7], this result is used to separate the fragments of Arithmetic introduced there: I n+1 (I n+1) and B * n+1 (I n+1). A basic result on n+1-induction rule is the following conservativeness theorem of C. Parsons (see [16] and 6.5): I n+1 is a n+2-conservative extension of

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Section: Ackermann's Functionssupporting

confidence: 52%

Section: ((3) ⇒ (1))mentioning

confidence: 99%

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On the quantifier complexity of ? n+1 (T)? induction

Cordón‐Franco¹,

Margarit²,

Lara‐Martín

2004

Archive for Mathematical Logic

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“…, and (M, s) |= (c). Then since q < p r by (6), and s ⊇ s r by assumption, (8) implies that g(r + 1) converges in M, which contradicts the choice of r and completes the proof of (c ).…”

Section: Formalized Second Recursion Theorem For Every Total Recursimentioning

confidence: 80%

“…(g) Prov Γ (y) abbreviates ∃x Prf Γ (x, y), which expresses "y is provable from Γ"; and Con(Γ) abbreviates ¬Prov Γ ( 0 = 1 ) , which expresses "Γ is consistent". 6 (h) Con(x, Γ) is the formula ∀v ≤ x ¬Prf Γ (v, 0 = 1 ), which expresses "there is no Γ-proof of 0 = 1 whose Gödel-number is at most x".…”

Section: Definitions and Conventionsmentioning

confidence: 99%

Marginalia on a Theorem of Woodin

Blanck¹,

Enayat²

2017

J. symb. log.

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Let $\left\langle {{W_n}:n \in \omega } \right\rangle$ be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index $e \in \omega$ (that depends on T) with the property that if${\cal M}$ is a countable model of T and for some${\cal M}$-finite set s, ${\cal M}$ satisfies ${W_e} \subseteq s$, then${\cal M}$ has an end extension${\cal N}$ that satisfies T + We = s.Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on ${\cal M}$ when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.

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Induction, minimization and collection for ? n+1 (T)?formulas

Margarit¹,

Lara‐Martín

2004

Archive for Mathematical Logic

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For a theory T, we study relationships among I∆n+1(T), L∆n+1(T) and B * ∆ n+1 (T). These theories are obtained restricting the schemes of induction, minimization and (a version of) collection to ∆n+1(T) formulas. We obtain conditions on T (T is an extension of B * ∆ n+1 (T) or ∆ n+1 (T) is closed (in T) under bounded quantification) under which I∆ n+1 (T) and L∆ n+1 (T) are equivalent.These conditions depend on ThΠ n+2 (T), the Πn+2-consequences of T. The first condition is connected with descriptions of Th Π n+2 (T) as IΣ n plus a class of nondecreasing total Πn-functions, and the second one is related with the equivalence between ∆n+1(T)formulas and bounded formulas (of a language extending the language of Arithmetic). This last property is closely tied to a general version of a well known theorem of R. Parikh.Using what we call Π n -envelopes we give uniform descriptions of the previous classes of nondecreasing total Πn-functions. Πn-envelopes are a generalization of envelopes (see [10]) and are closely related to indicators (see [12]). Finally, we study the hierarchy of theories I∆n+1(IΣm), m ≥ n, and prove a hierarchy theorem.

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A sharpened version of McAloon's theorem on initial segments of models of IΔ0

Cited by 8 publications

References 4 publications

On the quantifier complexity of ? n+1 (T)? induction

On the quantifier complexity of ? n+1 (T)? induction

Marginalia on a Theorem of Woodin

Induction, minimization and collection for ? n+1 (T)?formulas

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