Universe (mathematics)

A basic amount (including the universe ) called in mathematics a lot from all examined in a particular context objects. All considered in this context, quantities are then subsets of this basic set. In some cases, however, are not even considered all subsets of the universal set in return, for example in the case of a σ - algebra. In the logic or in the language sciences, the concept of basic quantity corresponds to the universe of discourse; In predicate logic, the definition set.

The use of basic quantities used to avoid paradoxes like Russell's antinomy quantity. Due to their suitable choice guarantees that set operations, such as averages and associations are defined and can now perform related only to more meaningful ( non-contradictory ) quantities.

Which objects can be included at all in the amount of solution to a given equation is crucially dependent on which base amount referred to in the equation.

In the case of an equation such as it is a statement form that is neither true nor false in itself. Only if one uses instead of concrete numbers x, is from the statement form, a statement that is either true or false. We are interested in solving an equation is usually the number that makes the equation a true statement. The one who came up with this equation, also makes for the solver of this equation have won yet another rule: you should be allowed to search only within the natural numbers by an object or a number that makes the equation a true statement. In other words: The basic amount for equation is as prescribed in this case. As a result of this limitation, it will not find any number which meets the equation. And so the solution set of the equation is empty.

But agreed to a different base set, one that contains the number, eg the set of integers or a more comprehensive set of numbers, then the above equation has a solution, namely. Therefore applies to the solution set.

The choice of a basic amount thus has a significant impact on whether an equation is solvable, and also on the number of elements of a possibly existing solution set. The same applies to inequalities and in general for statement forms, in which variables can occur.