scholarly journals A generalization of a conservativity theorem for classical versus intuitionistic arithmetic

2003 ◽  
Vol 50 (1) ◽  
pp. 41-46 ◽  
Author(s):  
Stefano Berardi
Keyword(s):  
Intuitionistic Arithmetic

Relativized realizability in intuitionistic arithmetic of all finite types

1978 ◽  
Vol 43 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Extension Theorem ◽  
General Notion ◽  
Conservative Extension ◽  
First Order ◽  
New Information ◽  
Reflection Theorem ◽  
Order Arithmetic ◽  
Special Case ◽  
Dependent Choice ◽  
Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

Intensional interpretations of functionals of finite type I

1967 ◽  
Vol 32 (2) ◽  
pp. 198-212 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Finite Type ◽  
Type I ◽  
Axiom Schema ◽  
Image Position ◽  
Intuitionistic Arithmetic

T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schemaof choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.

scholarly journals Implicational complexity in intuitionistic arithmetic

1981 ◽  
Vol 46 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Normal Form ◽  
Form Theorem ◽  
Natural Measure ◽  
Normal Form Theorem ◽  
Propositional Constants ◽  
Image Position ◽  
Classical Arithmetic ◽  
Intuitionistic Arithmetic

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].

Interpretations of Kleene's metamathematical predicate Γ ∣ a in intuitionistic arithmetic

1965 ◽  
Vol 30 (2) ◽  
pp. 140-154 ◽  
Author(s):  
T. Thacher Robinson
Keyword(s):  
Predicate Calculus ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Recursive Functions ◽  
Finite Sequences ◽  
Constructive Proofs ◽  
Necessary And Sufficient ◽  
Intuitionistic Arithmetic

Let Pp, Pd, and N be the intuitionistic systems of prepositional calculus, predicate calculus, and elementary arithmetic, respectively, described in [3].Kleene [4] introduces a metamathematical predicate Γ ∣ A for each of the systems Pp, Pd, and N, where Γ ranges over finite sequences of wffs, and A ranges over wffs, of that system. In the case of N, if Γ is consistent, then ‘ Γ ∣ A’ is essentially the result of deleting all references to recursive functions from the metamathematical predicate ‘A is realizable-(Γ ⊦)’ described in [3], pp. 502–503.Through use of this predicate, Kleene [4] obtains elegant constructive proofs of the following results for N:Metatheorem 0.1. If B ∨ C is a closed theorem of N, then ⊦ B or ⊦ C.Metatheorem 0.2. If (∃a)D(a) is a closed theorem of N, then there is a numeral n such that ⊦D(n).Metatheorem 0.3. If A is a closed wff of N, then A ∣ A is a necessary and sufficient condition that, for all closed B, C, (∃a)D(a) inN:(0.3.1) ⊦ A ⊃ B ∨ C implies ⊦ A ⊃ B or ⊦A ⊃ Cand(0.3.2) ⊦A ⊃ (∃)D(a) implies there is a numeral n such that ⊦ A ⊃ D(n).

Stewart Shapiro. Introduction—intensional mathematics and constructive mathematics. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, vol. 113, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 1–10. - Stewart Shapiro. Epistemic and intuitionistic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 11–46. - John Myhill. Intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 47–61. - Nicolas D. Goodman. A genuinely intensional set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 63–79. - Andrej Ščedrov. Extending Godel's modal interpretation to type theory and set theory. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 81–119. - Robert C. Flagg. Church's thesis is consistent with epistemic arithmetic. Intensional mathematics, edited by Stewart Shapiro, Studies in logic and the foundations of mathematics, pp. 121–172.

1991 ◽  
Vol 56 (4) ◽  
pp. 1496-1499 ◽  
Author(s):  
Craig A. Smoryński
Keyword(s):  
New York ◽  
Set Theory ◽  
Type Theory ◽  
Foundations Of Mathematics ◽  
Constructive Mathematics ◽  
Modal Interpretation ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Epistemic Arithmetic ◽  
Intuitionistic Arithmetic

Epistemic arithmetic is a conservative extension of intuitionistic arithmetic

1984 ◽  
Vol 49 (1) ◽  
pp. 192-203 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Propositional Calculus ◽  
Cut Elimination ◽  
Conservative Extension ◽  
First Order ◽  
Classical Mathematics ◽  
Order Arithmetic ◽  
Classical Propositional Calculus ◽  
Epistemic Arithmetic ◽  
Classical Arithmetic ◽  
Intuitionistic Arithmetic

Questions about the constructive or effective character of particular arguments arise in several areas of classical mathematics, such as in the theory of recursive functions and in numerical analysis. Some philosophers have advocated Lewis's S4 as the proper logic in which to formalize such epistemic notions. (The fundamental work on this is Hintikka [4].) Recently there have been studies of mathematical theories formalized with S4 as the underlying logic so that these epistemic notions can be expressed. (See Shapiro [7], Myhill [5], and Goodman [2]. The motivation for this work is discussed in Goodman [3].) The present paper is a contribution to the study of the simplest of these theories, namely first-order arithmetic as formalized in S4. Following Shapiro, we call this theory epistemic arithmetic (EA). More specifically, we show that EA is a conservative extension of Hey ting's arithmetic HA (ordinary first-order intuitionistic arithmetic). The question of whether EA is conservative over HA was raised but left open in Shapiro [7].The idea of our proof is as follows. We interpret EA in an infinitary propositional S4, pretty much as Tait, for example, interprets classical arithmetic in his infinitary classical propositional calculus in [8]. We then prove a cut-elimination theorem for this infinitary propositional S4. A suitable version of the cut-elimination theorem can be formalized in HA. For cut-free infinitary proofs, there is a reflection principle provable in HA. That is, we can prove in HA that if there is a cut-free proof of the interpretation of a sentence ϕ then ϕ is true. Combining these results shows that if the interpretation of ϕ is provable in EA, then ϕ is provable in HA.

scholarly journals Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

2021 ◽  
pp. 70-119
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Transcendental Number ◽  
Weierstrass Theorem ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

Lifschitz' realizability

1990 ◽  
Vol 55 (2) ◽  
pp. 805-821 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Axiomatic Characterization ◽  
Uniqueness Condition ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Continuous Application ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Intuitionistic Arithmetic

AbstractV. Lifschitz defined in 1979 a variant of realizability which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis. In this paper we describe an extension of intuitionistic arithmetic in which the soundness of Lifschitz' realizability can be proved, and we give an axiomatic characterization of the Lifschitz-realizable formulas relative to this extension. By a “q-variant” we obtain a new derived rule. We also show how to extend Lifschitz' realizability to second-order arithmetic. Finally we describe an analogous development for elementary analysis, with partial continuous application replacing partial recursive application.

scholarly journals Independence problems in subsystems of intuitionistic arithmetic

1971 ◽  
Vol 74 ◽  
pp. 448-456 ◽  
Author(s):  
D. van Dalen ◽  
C.E. Gordon
Keyword(s):  
Intuitionistic Arithmetic

Some models for intuitionistic finite type arithmetic with fan functional

1977 ◽  
Vol 42 (2) ◽  
pp. 194-202 ◽  
Author(s):  
A. S. Troelstra
Keyword(s):  
Finite Type ◽  
Church’S Thesis ◽  
Proper Understanding ◽  
New Model ◽  
Fan Theorem ◽  
Church's Thesis ◽  
Definition Of ◽  
Intuitionistic Arithmetic

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω+ MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide (since then the definition ofW12[T4, p. 594] is equivalent to the definition forW12in the case of ECF); but inELit is impossible to prove this (and under assumption of Church's thesis the two models differ).

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