In 1929, Mojżesz Presburger showed that the theory of natural numbers with addition and equality ( now called Presburger arithmetic in his honor ) is decidable and gave an algorithm that could determine if a given sentence in the language was true or false.

This means it is possible to effectively determine, for any sentence in the language of Presburger arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic.

In this language, the axioms of Presburger arithmetic are the universal closures of the following:

# Let P ( x ) be a first-order formula in the language of Presburger arithmetic with a free variable x ( and possibly other free variables ).

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

Initially, BASIC concentrated on supporting straightforward mathematical work, with matrix arithmetic support from its initial implementation as a batch language and full string functionality being added by 1965.

Since computers can make arithmetic calculations much faster and more accurately than humans, it was thought to be only a short matter of time before the technical details could be taken care of that would allow them the same remarkable capacity to process language.

The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers.

Rather, we say that a real a is definable in the language of arithmetic ( or arithmetical ) if its Dedekind cut can be defined as a predicate in that language ; that is, if there is a first-order formula φ in the language of arithmetic, with two free variables, such that

Sometimes " long double " is used for this in the C language family ( the C99 and C11 standards " IEC 60559 floating-point arithmetic extension-Annex F " recommend the 80-bit extended format to be provided as " long double " when available ), though " long double " may be a synonym for " double " or may stand for quadruple precision.

When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA < sub > 0 </ sub > ( a second-order variant of Peano arithmetic restricted to induction over Σ < sup > 0 </ sup >< sub style =" margin-left :- 0. 6em "> 1 </ sub > formulas ).

The language contains ordinary arithmetic and comparison operators, and unconditional, conditional, and assigned jumps.

Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967.

However, the spoken language uses both arithmetic and geometric numerals.

The Ambi programming language uses Polish Notation for arithmetic operations and program construction.

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

The full complement of arithmetic and binary operators is defined on vector types so that the normal C expression language can be used to manipulate vector variables.

While the language provides simple matrix operations such as multiplication, the Scilab package also provides a library of high-level operations such as correlation and complex multidimensional arithmetic.

The alphabet in the Polish language contains 32 letters.

For most speakers ( even native speakers ), this form of the language is generally difficult to understand, as it contains many highly specialised terms for diplomatic, internal, official, and military matters.

The XML Schema language, sometimes used in connection with representing data for storage in computers, contains built-in primitive datatypes, date and dateTime, which do not allow a year zero, and designate years BC as negative numbers.

In a language with records and references, binary trees are typically constructed by having a tree node structure which contains some data and references to its left child and its right child.

In languages with tagged unions such as ML, a tree node is often a tagged union of two types of nodes, one of which is a 3-tuple of data, left child, and right child, and the other of which is a " leaf " node, which contains no data and functions much like the null value in a language with pointers.

A formal grammar defines ( or generates ) a formal language, which is a ( usually infinite ) set of finite-length sequences of symbols ( i. e. strings ) that may be constructed by applying production rules to another sequence of symbols which initially contains just the start symbol.

A scene file contains objects in a strictly defined language or data structure ; it would contain geometry, viewpoint, texture, lighting, and shading information as a description of the virtual scene.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

A 1989 recording of Enya singing " Oíche Chiúin ", an Irish language version of " Silent Night ", has been reissued at least twice: on The Christmas EP ( which otherwise contains several non-holiday related previously issued recordings by Enya ) and the 1997 edition of the charity album A Very Special Christmas.

Halakhot Gedolot (" Great Law Book "), by R. Simeon Kayyara, published two generations later, contains extensive additional material, mainly from Responsa and Monographs of the Geonim, and is presented in a form that is closer to the original Talmud language and structure.

Some examples of leet include B1FF and n00b, a term for the stereotypical newbie ; the l33t programming language ; and the webcomic Megatokyo, which contains characters who speak leet.

This work deals mainly with language, beginning with the reflection on artistic expression in The Structure of Behavior-which contains a passage on El Greco ( p. 203ff ) that prefigures the remarks that he develops in " Cézanne's Doubt " ( 1945 ) and follows the discussion in Phenomenology of Perception.

This volume contains many of his early papers, in which he argues for a deep connection between language and social structure.

Flash contains an object-oriented language called ActionScript and supports automation via the JavaScript Flash language ( JSFL ).

Conversely, Scheme contains multiple coherent subsets that suffice to construct the rest of the language as library macros, and so the language designers do not even bother to say which portions of the language must be implemented as language constructs, and which must be implemented as parts of a library.

For example, Java is both an object-oriented language ( because it encourages object-oriented organization ) and a concurrent language ( because it contains built-in constructs for running multiple threads in parallel ).

In the theory of formal languages in computability theory, a pumping lemma or pumping argument states that, for a particular language to be a member of a language class, any sufficiently long string in the language contains a section, or sections, that can be removed, or repeated any number of times, with the resulting string remaining in that language.

The language of VBScript is modelled on Visual Basic, and therefore can be reviewed using similar categories: procedures, control structures, constants, variables, user interaction, array handling, date / time functions, error handling, mathematical functions, objects, regular expressions, string manipulation, and so on.

Assembly language source code generally permits the use of constants and programmer comments.

The type of logic called propositional, sentential, or statement logic includes only operators and propositional constants as symbols in its language.

The propositions in this language are propositional constants, which are considered atomic propositions, and composite propositions, which are composed by recursively applying operators to propositions.

Assembly language may itself be regarded as a higher level ( but often still one-to-one if used without macros ) representation of machine code, as it supports concepts such as constants and ( limited ) expressions, sometimes even variables, procedures, and data structures.

:" I come to semantic investigations with a preference for homophonic theories ; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ... I would prefer a theory ... over a theory which is only able to deal with of the form " all A's are B's " by " discovering " hidden logical constants ...

We are, in classical language, looking at invariants of n-ary r-ics, where n is the dimension of V. ( This is not the same as finding invariants of GL ( V ) on S ( V ); this is an uninteresting problem as the only such invariants are constants.

For instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language ( e. g. parentheses ).

The C # language also permits the definition to include properties owned by the object, which are treated as methods with syntactic sugar, but no constants ( in contrast to Java ), constructors, destructors, fields, nested types, or operators.

Pthreads defines a set of C programming language types, functions and constants.

All constants can be assigned to variables, which have the same form as in ALGOL, but for which no fixed types are specified: Euler is a dynamically typed programming language.

An expression in a programming language is a combination of explicit values, constants, variables, operators, and functions that are interpreted according to the particular rules of precedence and of association for a particular programming language, which computes and then produces ( returns, in a stateful environment ) another value.

In the language consisting of the binary functions + and ×, binary relation <, and constants 0 and 1, the structure ( Q, +, ×, <, 0, 1 ) is a substructure of ( R, +, ×, <, 0, 1 ).

Let be a theory obtained from by extending its language with new constants

Then is a conservative extension of, which means that the theory has the same set of theorems in the original language ( i. e., without constants ) as the theory.

A first-order theory T has quantifier elimination if and only if for any two models B and C of T and for any common substructure A of B and C, B and C are elementarily equivalent in the language of T augmented with constants from A.

The signature σ of a language is a triple < O, F, P > consisting of the alphabet of constants O, the function symbols F, and the predicates P. The Herbrand base of a signature σ consists of all ground atoms of σ: of all formulas of the form R ( t < sub > 1 </ sub >, …, t < sub > n </ sub >), where t < sub > 1 </ sub >, …, t < sub > n </ sub > are terms containing no variables ( i. e. elements of the Herbrand universe ) and R is an n-ary relation symbol ( i. e. predicate ).

* $ is used for defining hexadecimal constants in Pascal-like languages such as Delphi, and in some variants of assembly language.