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In mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [ 1 ] Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics: Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes: [6] [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive ...
Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.
Mathematics in the twentieth century evolved into a network of axiomatic formal systems. This was, in considerable part, influenced by the example Hilbert set in the Grundlagen .
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical ...
In mathematics and logic, an axiomatic system is any set of primitive notions and axioms to logically derive theorems.A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems.
Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero. [4] For example, it notes that the sum of a positive number and a negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive.
This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an undefined output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an ordering relation, in which the "undefined result" is the least element.