If it is not satisfiable, then every sentence in the associated language is provable (since there is no model in which it might be false). A satisfiable system is necessarily consistent. a sentence and its negation) is consistent. A theory in which it is impossible to derive a contradiction (i.e. a model in which all sentences of the theory are true) is said to be satisfiable. Gödel's Completeness Theorem states that, for a certain kind ( first-order, to be precise) of a language, a sentence can be proved in a theory if and only if it is true in every model of that theory.Ī theory that has a model (i.e. Thus a sentence may or may not be a logical consequence of a theory and may or may not be true in one or more models of that theory. (A proof is a deduction in a finite number of steps.) In a model, a sentence may be true or not. In other words, it may be inquired whether a sentence has a proof in a theory. For a theory and a sentence in a certain language, a meaningful question to ask is whether that sentence can be deduced in the theory following the rules of the language. For example, a language that includes some logical symbols along with variables and a symbol for a single binary operation may be used to describe Group Theory, for which various groups serve as models. Mathematical structures are models or interpretations of such theories. A theory (in a certain language) is a collection of sentences formed in that language.

Below is just a short, introductory outline of the ideas involved.Ī formal language includes a set of rules for forming valid sentences. These concepts deserve a more extended exposition. Mathematical logic is a study of formal languages that are used to describe mathematical structures. Įssential to the understanding of Robinson's approach is mathematical logic and theory of models. Henkin, and the German/American Kurt Gödel. Skolem, the Russian Anatoli Maltsev, the American Leon A. Robinson's construction of what became known as the Non-standard Analysis came on the heels of the work done earlier by the Norwegian Thoralf A. Quite remarkably in the 20 th century, Abraham Robinson from the Hebrew University of Jerusalem, Israel, while on leave at the Institute for Advanced Study in Princeton during the 1959-1960 academic year, managed to create a theory of infinitesimals on a superbly logical foundation that depended less on algebra and arithmetic but more on the understanding of that side of mathematics that makes use of formal languages to describe theories and their models.

The place of 'clear, precise definitions' is in Analysis, which is a foundational part of Calculus and is being studied by mathematically inclined students.) This is how it has been used by mathematicians and physists even before, and certainly since, it was formulated as such by Newton and Leibniz. Calculus is supposed to be a tool chest of methods for solving certain kinds of problems. The new approach wasn't easy, though, as students who have had to learn his 'epsilon-delta' approach to limits will still testify.' I believe that nothing is easy to an unprepared mind. For example, wrote about Weierstrass, 'His clear, precise definitions removed any trace of mystery or geometric intuition from calculus, putting it all on a logical foundation that depended only on algebra and arithmetic. (As an aside, Weierstrass' definition is often singled out by mathematics educators as a piece of calculus especially difficult for the beginning students. Starting with Newton and Leibniz in the 17 th century, practically all great mathematicians tried unsuccessfully to justify the employment of infinitesimals till in the 19 th century the infinitesimals were finally banished from mathematics and replaced with Weierstrass' ε-δ definition of the limit. The early history of Calculus is the story of infinitesimals.

Problems of infinitesimals