Flagg realizability in arithmetic

1986 ◽  
Vol 51 (2) ◽  
pp. 387-392 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Church’S Thesis ◽  
First Order ◽  
Suitable Form ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Epistemic Arithmetic

Epistemic arithmetic—that is, first-order arithmetic with S4 as the underlying logic—was introduced by Shapiro in [7] and independently by Reinhardt in [6]. Shapiro showed that intuitionistic first-order arithmetic HA can be embedded in epistemic arithmetic EA. Moreover he showed that some of the basic proof-theoretic facts about HA, such as the existence and disjunction properties, can be extended to EA. In [3] we showed that the interpretation of HA in EA is faithful. (G.E. Mine has independently also proved this theorem.) Finally, in [2], Flagg showed that a suitable form of Church's thesis is consistent with EA. (Carlson [1] has announced another proof of this result.) Flagg's argument involves an ingenious realizability notion for EA which, as it stands, is not very perspicuous. The purpose of the present paper is to give a more transparent treatment of Flagg realizability. We obtain a new version of Flagg's proof of the consistency of Church's thesis with EA. Our main new result is that, in a sense to be made precise below, Flagg realizability coincides on HA embedded in EA with Kleene's 1945 realizability (e.g. see [5, pp. 501–516]). Thus it turns out once more that methods and results proved for EA can be viewed as extensions or generalizations of well-known methods and results for HA.

scholarly journals REALIZABILITY SEMANTICS FOR QUANTIFIED MODAL LOGIC: GENERALIZING FLAGG’S 1985 CONSTRUCTION

2016 ◽  
Vol 9 (4) ◽  
pp. 752-809 ◽  
Author(s):  
BENJAMIN G. RIN ◽  
SEAN WALSH
Keyword(s):  
Modal Logic ◽  
Graph Model ◽  
Church’S Thesis ◽  
Quantified Modal Logic ◽  
First Order ◽  
Continuity Principle ◽  
Generalized Continuity ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
The Stability

AbstractA semantics for quantified modal logic is presented that is based on Kleene’s notion of realizability. This semantics generalizes Flagg’s 1985 construction of a model of a modal version of Church’s Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church’s Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra’s generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott’s graph model (for the untyped lambda calculus) which witnesses the failure of the stability of nonidentity.

Extension of Lifschitz' realizability to higher order arithmetic, and a solution to a problem of F. Richman

1991 ◽  
Vol 56 (3) ◽  
pp. 964-973 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Image Position

AbstractF. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH:and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP.In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

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.

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.

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.

Proving Church's Thesis

2000 ◽  
Vol 8 (3) ◽  
pp. 244-258 ◽  
Author(s):  
ROBERT BLACK
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

scholarly journals Reflections on Church's thesis.

1987 ◽  
Vol 28 (4) ◽  
pp. 490-498 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

On the role of Ramsey quantifiers in first order arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 423-435 ◽  
Author(s):  
James H. Schmerl ◽  
Stephen G. Simpson
Keyword(s):  
Formal System ◽  
Intended Meaning ◽  
Completeness Proof ◽  
First Order ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Witness Set ◽  
Order Arithmetic ◽  
Infinite Set

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X.Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12].After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

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