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TeX is a mathematically oriented typesetting system that was created in 1978 by Donald Knuth. It is widely used in mathematics, through its extension called LaTeX, and is a de facto standard. (The above expression is written in LaTeX.) More recently, another approach for mathematical typesetting is provided by MathML. However, it is not well ...
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics , logic , and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation.
A dictionary writing system (DWS), or dictionary production/publishing system (DPS) is software for writing and producing a dictionary, glossary, vocabulary, or thesaurus. It may include an editor, a database , a web interface for collaborative work, and various management tools.
A key fact is that one can translate sentences from the language of the interpreted structures to the language of the original structure. Thus one can show that if a structure M {\displaystyle {\mathcal {M}}} interprets another whose theory is undecidable, then M {\displaystyle {\mathcal {M}}} itself is undecidable.
A formula A in a language is satisfiable if it is true for some interpretation of . A formula A of the language of arithmetic is decidable if it represents a decidable set , i.e. if there is an effective method which, given a substitution of the free variables of A , says that either the resulting instance of A is provable or its negation is.
Any language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be a natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language , that is, the ...
Joseph M. Williams (18 August 1933 in Cleveland, Ohio – 22 February 2008 in South Haven, Michigan) was a professor in the Department of English Language and Literature at the University of Chicago where he promoted clarity in writing for many years. He authored several books on language and writing.
The consequence of these features is that a mathematical text is generally not understandable without some prerequisite knowledge. For example, the sentence "a free module is a module that has a basis" is perfectly correct, although it appears only as a grammatically correct nonsense, when one does not know the definitions of basis, module, and free module.