Models of intuitionistic TT and NF

1995 ◽  
Vol 60 (2) ◽  
pp. 640-653 ◽  
Author(s):  
Daniel Dzierzgowski
Keyword(s):  
Classical Theory ◽  
Simple Technique ◽  
Peano Arithmetic ◽  
Simple Theory ◽  
Kripke Models ◽  
Finite Sets ◽  
Heyting Arithmetic ◽  
Intuitionistic Theory ◽  
Excluded Middle ◽  
Intuitionistic Version

AbstractLet us define the intuitionistic part of a classical theory T as the intuitionistic theory whose proper axioms are identical with the proper axioms of T. For example, Heyting arithmetic HA is the intuitionistic part of classical Peano arithmetic PA.It's a well-known fact, proved by Heyting and Myhill, that ZF is identical with its intuitionistic part.In this paper, we mainly prove that TT, Russell's Simple Theory of Types, and NF, Quine's “New Foundations,” are not equal to their intuitionistic part. So, an intuitionistic version of TT or NF seems more naturally definable than an intuitionistic version of ZF.In the first section, we present a simple technique to build Kripke models of the intuitionistic part of TT (with short examples showing bad properties of finite sets if they are defined in the usual classical way).In the remaining sections, we show how models of intuitionistic NF2 and NF can be obtained from well-chosen classical ones. In these models, the excluded middle will not be satisfied for some non-stratified sentences.

Forcing and satisfaction in Kripke models of intuitionistic arithmetic

2018 ◽  
Vol 27 (5) ◽  
pp. 659-670
Author(s):  
Maryam Abiri ◽  
Morteza Moniri ◽  
Mostafa Zaare
Keyword(s):  
Predicate Logic ◽  
Kripke Model ◽  
Peano Arithmetic ◽  
Kripke Models ◽  
Classical Structure ◽  
First Order ◽  
Heyting Arithmetic ◽  
Intuitionistic Arithmetic

Abstract We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property is an $\mathsf{E}$-formula. Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic.

scholarly journals The Oxygen 1.13 μm Fluorescence Line of SN1987A: a Diagnostic for the Ejecta of Hydrogen-Rich Supernovae

1996 ◽  
Vol 145 ◽  
pp. 235-240
Author(s):  
Ernesto Oliva
Keyword(s):  
Relative Abundance ◽  
Classical Theory ◽  
Filling Factor ◽  
Simple Theory ◽  
Maximum Distance ◽  
Fluorescence Line ◽  
Hydrogen Lines

The bright O I λ11287 line observed in SN1987A is produced by the Bowen fluorescence with Lyβ and comes from regions that lie within a Sobolev length (δR ∼ 10-3RSN, the maximum distance over which fluorescence can work) from hydrogen rich gas ionized by the 56Co decay. Its strength relative to hydrogen lines (e.g. Brγ) depends on the O/H relative abundance in the ‘fluorescent region’ and on the density (i.e. the filling factor) of the gas. The observed evolution of λ11287 can be successfully understood using a relatively simple theory which takes into account the effects of transfer in the O I lines and is the generalization of the classical theory of Bowen fluorescence.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

The undecidability of intuitionistic theories of algebraically closed fields and real closed fields

1973 ◽  
Vol 38 (1) ◽  
pp. 86-92 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Classical Theory ◽  
Linear Order ◽  
Abelian Groups ◽  
Classical Model ◽  
Sufficient Conditions ◽  
Interesting Case ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Intuitionistic Theory ◽  
Algebraically Closed Fields

Let T be a set of axioms for a classical theory TC (e.g. abelian groups, linear order, unary function, algebraically closed fields, etc.). Suppose we regard T as a set of axioms for an intuitionistic theory TH (more precisely, we regard T as axioms in Heyting's predicate calculus HPC).Question. Is TH decidable (or, more generally, if X is any intermediate logic, is TX decidable)? In [1] we gave sufficient conditions for the undecidability of TH. These conditions depend on the formulas of T (different axiomatization of the same TC may give rise to different TH) and on the classical model theoretic properties of TC (the method did not work for model complete theories, e.g. those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories TH for a classically model complete TC remained open. An undecidability result in this direction, for dense linear order was obtained by Smorynski [4]. The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.One more remark before we start. There are several possible formulations for an intuitionistic theory of, e.g. fields, that correspond to several possible axiomatizations of the classical theory. Other formulations may be given in terms of the apartness relation, such as the one for fields given by Heyting [5]. The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5].

scholarly journals Kripke models and the intuitionistic theory of species

1976 ◽  
Vol 9 (1-2) ◽  
pp. 157-186 ◽  
Author(s):  
D.H.J. de Jongh ◽  
C. Smorynski
Keyword(s):  
Kripke Models ◽  
Intuitionistic Theory

scholarly journals NNIL-formulas revisited: Universal models and finite model property

2020 ◽  
Author(s):  
Julia Ilin ◽  
Dick de Jongh ◽  
Fan Yang
Keyword(s):  
Direct Proof ◽  
Finite Model ◽  
Kripke Models ◽  
Model Property ◽  
Finite Model Property ◽  
Universal Models ◽  
Heyting Arithmetic ◽  
First Results

Abstract NNIL-formulas, introduced by Visser in 1983–1984 in a study of $\varSigma _1$-subsitutions in Heyting arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The first results about these formulas were obtained in a paper of 1995 by Visser et al. In particular, it was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. Recently, Bezhanishvili and de Jongh observed that NNIL-formulas are also reflected by the colour-preserving monotonic maps of Kripke models. In the present paper, we first show how this observation leads to the conclusion that NNIL-formulas are preserved by arbitrary substructures not necessarily satisfying the topo-subframe condition. Then, we apply it to construct universal models for NNIL. It follows from the properties of these universal models that NNIL-formulas are also exactly the formulas that are reflected by colour-preserving monotonic maps. By using the method developed in constructing the universal models, we give a new direct proof that the logics axiomatized by NNIL-axioms have the finite model property.

Kripke models of certain subtheories of Heyting Arithmetic

2017 ◽  
pp. 136-142
Author(s):  
Tomasz Połacik
Keyword(s):  
Kripke Models ◽  
Heyting Arithmetic

A new method for establishing conservativity of classical systems over their intuitionistic version

1999 ◽  
Vol 9 (4) ◽  
pp. 323-333 ◽  
Author(s):  
THIERRY COQUAND ◽  
MARTIN HOFMANN
Keyword(s):  
Polynomial Time ◽  
New Method ◽  
Bounded Arithmetic ◽  
Kripke Models ◽  
Classical Systems ◽  
Skolem Functions ◽  
Intuitionistic Version

We use a syntactical notion of Kripke models to obtain interpretations of subsystems of arithmetic in their intuitionistic counterparts. This yields, in particular, a new proof of Buss' result that the Skolem functions of Bounded Arithmetic are polynomial time computable.

scholarly journals The quantum strategy of completeness: On the self-foundation of mathematics

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Mechanics ◽  
Hidden Variables ◽  
Peano Arithmetic ◽  
The Self ◽  
Transfinite Induction ◽  
Experimental Science ◽  
Heyting Arithmetic ◽  
Philosophical Background ◽  
Concept Of Information ◽  
The Cost

Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation.

From Second-Order Heyting Arithmetic to Second-Order Peano Arithmetic

1998 ◽  
pp. 434-442
Author(s):  
Peter Fletcher
Keyword(s):  
Peano Arithmetic ◽  
Second Order ◽  
Heyting Arithmetic
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