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At its core, this proof is non-constructive because it relies on the statement "Either q is rational or it is irrational"—an instance of the law of excluded middle, which is not valid within a constructive proof. The non-constructive proof does not construct an example a and b; it merely gives a number of possibilities (in this case, two ...
In the U.S. legal system, service of process is the procedure by which a party to a lawsuit gives an appropriate notice of initial legal action to another party (such as a defendant), court, or administrative body in an effort to exercise jurisdiction over that person so as to force that person to respond to the proceeding in a court, body, or other tribunal.
For example, any theorem of classical propositional logic of the form has a proof consisting of an intuitionistic proof of followed by one application of double-negation elimination. Intuitionistic logic can thus be seen as a means of extending classical logic with constructive semantics.
In classical real analysis, one way to define a real number is as an equivalence class of Cauchy sequences of rational numbers.. In constructive mathematics, one way to construct a real number is as a function ƒ that takes a positive integer and outputs a rational ƒ(n), together with a function g that takes a positive integer n and outputs a positive integer g(n) such that
The above proof is an example of a non-constructive proof disallowed by intuitionists: The proof is non-constructive because it doesn't give specific numbers a {\\displaystyle a} and b {\\displaystyle b} that satisfy the theorem but only two separate possibilities, one of which must work.
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that a b {\displaystyle a^{b}} is a rational number .
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...