scholarly journals Minimal models of Heyting arithmetic

1997 ◽  
Vol 62 (4) ◽  
pp. 1448-1460 ◽  
Author(s):  
Ieke Moerdijk ◽  
Erik Palmgren
Keyword(s):  
Type Theory ◽  
Geometric Theory ◽  
Minimal Models ◽  
Nonstandard Model ◽  
Heyting Arithmetic ◽  
Suitable Site ◽  
Definition Of ◽  
A Site ◽  
Effective Proof ◽  
Intuitionistic Arithmetic

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be called HAI and HAIω respectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

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.

General models and extensionality

1972 ◽  
Vol 37 (2) ◽  
pp. 395-397 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Axiom Schema ◽  
Nonstandard Model ◽  
Equality Relation ◽  
Definition Of

It is well known that equality is definable in type theory. Thus, in the language of [2], the equality relation between elements of type α is definable as , i.e., xα = yα iff every set which contains xα also contains yα. However, in a nonstandard model of type theory, the sets may be so sparse that the wff above does not denote the true equality relation. We shall use this observation to construct a general model in the sense of [2] in which the Axiom of Extensionality is not valid. Thus Theorem 2 of [2] is technically incorrect. However, it is easy to remedy the situation by slightly modifying the definition of general model.Our construction will show that the Axiom Schema of Extensionality is independent even if one takes as an axiom schema.We shall assume familiarity with, and use the notation of, [2] and §§2–3 of [1].

MONARCHS, TRAVELLERS AND EMPIRE IN THE PACIFIC'S AGE OF REVOLUTIONS

2020 ◽  
Vol 30 ◽  
pp. 77-96
Author(s):  
Sujit Sivasundaram
Keyword(s):  
Pacific Islanders ◽  
Modern World ◽  
Global History ◽  
The Past ◽  
The Pacific ◽  
Cultural Terms ◽  
The Pacific Ocean ◽  
Definition Of ◽  
The Uk ◽  
A Site

AbstractThe Pacific has often been invisible in global histories written in the UK. Yet it has consistently been a site for contemplating the past and the future, even among Britons cast on its shores. In this lecture, I reconsider a critical moment of globalisation and empire, the ‘age of revolutions’ at the end of the eighteenth century and the start of the nineteenth century, by journeying with European voyagers to the Pacific Ocean. The lecture will point to what this age meant for Pacific islanders, in social, political and cultural terms. It works with a definition of the Pacific's age of revolutions as a surge of indigeneity met by a counter-revolutionary imperialism. What was involved in undertaking a European voyage changed in this era, even as one important expedition was interrupted by news from revolutionary Europe. Yet more fundamentally vocabularies and practices of monarchy were consolidated by islanders across the Pacific. This was followed by the outworkings of counter-revolutionary imperialism through agreements of alliance and alleged cessation. Such an argument allows me, for instance, to place the 1806 wreck of the Port-au-Prince within the Pacific's age of revolutions. This was an English ship used to raid French and Spanish targets in the Pacific, but which was stripped of its guns, iron, gunpowder and carronades by Tongans. To chart the trajectory from revolution and islander agency on to violence and empire is to appreciate the unsettled paths that gave rise to our modern world. This view foregrounds people who inhabited and travelled through the earth's oceanic frontiers. It is a global history from a specific place in the oceanic south, on the opposite side of the planet to Europe.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

DELIGHTFUL COXCOMBS TO INDUSTRIOUS MEN: FASHIONABLE POLITICS IN CECIL AND PENDENNIS

2002 ◽  
Vol 30 (1) ◽  
pp. 289-304 ◽  
Author(s):  
Claire Nicolay
Keyword(s):  
Nineteenth Century ◽  
Eighteenth Century ◽  
Middle Class ◽  
Upward Mobility ◽  
Late Nineteenth Century ◽  
Class A ◽  
Late Nineteenth ◽  
Definition Of ◽  
A Site ◽  
Trade Office

THOMAS CARLYLE’S CONTEMPTUOUS DESCRIPTION of the dandy as “a Clothes-wearing Man, a Man whose trade, office, and existence consists in the wearing of Clothes” (313) has survived as the best-known definition of dandyism, which is generally equated with the foppery of eighteenth-century beaux and late nineteenth-century aesthetes. Actually, however, George Brummell (1778–1840), the primary architect of dandyism, developed not only a style of dress, but also a mode of behavior and style of wit that opposed ostentation. Brummell insisted that he was completely self-made, and his audacious self-transformation served as an example for both parvenus and dissatisfied nobles: the bourgeois might achieve upward mobility by distinguishing himself from his peers, and the noble could bolster his faltering status while retaining illusions of exclusivity. Aristocrats like Byron, Bulwer, and Wellington might effortlessly cultivate themselves and indulge their taste for luxury, while at the same time ambitious social climbers like Brummell, Disraeli, and Dickens might employ the codes of dandyism in order to establish places for themselves in the urban world. Thus, dandyism served as a nexus for the declining aristocratic elite and the rising middle class, a site where each was transformed by the dialectic interplay of aristocratic and individualistic ideals.

scholarly journals Definition of essential sequences and functional equivalence of elements downstream of the adenovirus E2A and the early simian virus 40 polyadenylation sites.

1985 ◽  
Vol 5 (11) ◽  
pp. 2975-2983 ◽  
Author(s):  
R P Hart ◽  
M A McDevitt ◽  
H Ali ◽  
J R Nevins
Keyword(s):  
Essential Element ◽  
Simian Virus 40 ◽  
Simian Virus ◽  
Cleavage Sites ◽  
Multiple Sequence ◽  
Polyadenylic Acid ◽  
Sequence Elements ◽  
Cleavage And Polyadenylation ◽  
Definition Of ◽  
A Site

In addition to the highly conserved AATAAA sequence, there is a requirement for specific sequences downstream of polyadenylic acid [poly(A)] cleavage sites to generate correct mRNA 3' termini. Previous experiments demonstrated that 35 nucleotides downstream of the E2A poly(A) site were sufficient but 20 nucleotides were not. The construction and assay of bidirectional deletion mutants in the adenovirus E2A poly(A) site indicates that there may be redundant multiple sequence elements that affect poly(A) site usage. Sequences between the poly(A) site and 31 nucleotides downstream were not essential for efficient cleavage. Further deletion downstream (3' to +31) abolished efficient cleavage in certain constructions but not all. Between +20 and +38 the sequence T(A/G)TTTTT was duplicated. Function was retained when one copy of the sequence was present, suggesting that this sequence represents an essential element. There may also be additional sequences distal to +43 that can function. To establish common features of poly(A) sites, we also analyzed the early simian virus 40 (SV40) poly(A) site for essential sequences. An SV40 poly(A) site deletion that retained 18 nucleotides downstream of the cleavage site was fully functional while one that retained 5 nucleotides downstream was not, thus defining sequences required for cleavage. Comparison of the SV40 sequences with those from E2A did not reveal significant homologies. Nevertheless, normal cleavage and polyadenylation could be restored at the early SV40 poly(A) site by the addition of downstream sequences from the adenovirus E2A poly(A) site to the SV40 +5 mutant. The same sequences that were required in the E2A site for efficient cleavage also restored activity to the SV40 poly(A) site.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

Gorilla Warfare

2018 ◽  
Author(s):  
Sam Dubal
Keyword(s):  
Guerrilla Warfare ◽  
Colonial Era ◽  
Definition Of ◽  
A Site

This chapter investigates how humanity was constructed against animality through the space of the “bush” (lum). Whereas Acholi civilians and others had come to see the lum as a wild, dangerous, and evil space of animals, the rebels occupied the lum and gave it a different meaning. In what became a contestation over an “anthropomoral geography,” the LRA collapsed an analytic separating animality and humanity, unsettling a spatio-moral definition of humanity against animality. They saw their rebellion as gorilla rather than guerrilla warfare. Instead of reinforcing a colonial-era notion of the lum, the LRA found the lum to be a site of life, sacredness, and development. In doing so, they dissolved some of the spatio-moral infrastructure of “humanity” itself.

scholarly journals Transporting functions across ornaments

2014 ◽  
Vol 24 (2-3) ◽  
pp. 316-383 ◽  
Author(s):  
PIERRE-ÉVARISTE DAGAND ◽  
CONOR McBRIDE
Keyword(s):  
Type Theory ◽  
Structured Data ◽  
Dependent Types ◽  
Natural Numbers ◽  
Code Reuse ◽  
Domain Specific ◽  
Definition Of ◽  
The Relationship ◽  
Do So ◽  
Structural Invariants

AbstractProgramming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: We can finally write correct-by-construction software. However, this extreme accuracy is also a curse: A datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any attempt to reuse code across similarly structured data. In this paper, we capitalise on the structural invariants of datatypes. To do so, we first adapt the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: The users can ask the definition of addition to be lifted to lists and they will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in the type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

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