Fragments of $HA$ based on $\Sigma_1$ -induction

Wehmeier, Kai F.

doi:10.1007/s001530050081

Cited by 30 publications

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“…Recall Wehmeier's result, iΠ 1 exp, where exp is the Π 2 sentence which says the exponentiation function is total. His proof is based on constructing a two-node Kripke model of iΠ 1 such that its root is not a model of exp, see [W1,Lemma 10]. Here we prove a stronger independence result.…”

Section: An Intuitionistic Theory Tmentioning

confidence: 77%

Intuitionistic weak arithmetic

Moniri

2003

Archive for Mathematical Logic

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We construct ω-framed Kripke models of i∀ 1 and iΠ 1 non of whose worlds satisfies ∀x∃y(x = 2y ∨ x = 2y + 1) and ∀x, y∃zExp(x, y, z) respectively. This will enable us to show that i∀ 1 does not prove ¬¬∀x∃y(x = 2y ∨ x = 2y + 1) and iΠ 1 does not prove ¬¬∀x, y∃zExp(x, y, z). Therefore, i∀ 1 ¬¬lop and iΠ 1 ¬¬iΣ 1 . We also prove that HA lΣ 1 and present some remarks about iΠ 2 .2000 Mathematics Subject Classification: 03F30, 03F55, 03H15.Key words and phrases: Fragments of Heyting Arithmetic, Kripke Models, exp. We fix the language L = {+, ·, <, 0, 1} of arithmetic throughout the paper. PreliminariesBy open formulas we mean quantifier-free formulas. (∃x ≤ t)ϕ is an abbreviation for ∃x(x ≤ t ∧ ϕ) and (∀x ≤ t)ϕ is an abbreviation for ∀x(x ≤ t → ϕ), where t is a term not involving x. A formula is bounded if all quantifiers occurring in it are bounded, i.e., occur in a context as above. Σ 0 , Π 0 or ∆ 0 -formulas are bounded formulas. For n ≥ 0, Σ n+1 -formulas have the form (∃x)ϕ where ϕ in Π n , Π n+1 -formulas have the form (∀x)ϕ where ϕ in Σ n .The hierarchy of ∀ n -formulas and of ∃ n -formulas are defined similarly by changing bounded formulas to open formulas.Heything arithmetic HA and its fragments (P A − ) i , iop(= iopen), lop(= lopen) and i∆ 0 are the intuitionistic counterparts of first order Peano Arithmetic P A and its fragments P A − , Iop(= Iopen), Lop(= Lopen) and I∆ 0 . More generally for any set Γ of formulas we will use notations such as iΓ and lΓ in the same manner.

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Section: An Intuitionistic Theory Tmentioning

confidence: 77%

Intuitionistic weak arithmetic

Moniri

2003

Archive for Mathematical Logic

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“…T h e o r e m (WEHMEIER [15]). Suppose that 1x1 t-i E 3 y A ( f , y ) , where A is an arbitrary formula of the language of L whose free variables are among c and y.…”

Section: Harnik's Results Can Be Reformulated Thusmentioning

confidence: 99%

Extracting Algorithms from Intuitionistic Proofs

Ferreira

Marques²

1998

Mathematical Logic Qtrly

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This paper presents a new method -which does not rely on the cut-elimination theorem -for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory Si, and we briefly sketch it for the other levels of bounded arithmetic and for the theory I&. Mathematics Subject Classification: 03F30, 03F55, 03D15.

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“…Wehmeier [29] used Kleene's realizability to characterize provably total functions of ¦ ½ : "each of them is primitive recursive". This had (already) been proven by Damnjanovic [7] using so-called strictly primitive recursive realizability (see also [9]).…”

Section: Introductionmentioning

confidence: 99%

Provably total functions of Basic Arithmetic

Salehi

2003

Mathematical Logic Qtrly

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It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives.

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Fragments of $HA$ based on $\Sigma_1$ -induction

Cited by 30 publications

References 0 publications

Intuitionistic weak arithmetic

Intuitionistic weak arithmetic

Extracting Algorithms from Intuitionistic Proofs

Provably total functions of Basic Arithmetic

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