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This contrasts with ''classical analysis'', which (in this context) simply means analysis done according to the (ordinary) principles of Classical Mathematics . Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to Separable Space s; also, some theorems may need to be approached by Approximation s. Furthermore, many classical theorems can be stated in ways that are Logically Equivalent according to Classical Logic , but not all of these forms will be valid in constructive analysis, which uses Intuitionistic Logic . EXAMPLES The intermediate value theorem For a simple example, consider the Intermediate Value Theorem (IVT). In classical analysis, IVT says that, given any Continuous Function ''f'' from a Closed Interval {Link without Title} to the Real Line ''R'', if ''f''(''a'') is Negative while ''f''(''b'') is Positive , then there exists a Real Number ''c'' in the interval such that ''f''(''c'') is exactly Zero . In constructive analysis, this does not hold, because the constructive interpretation of Existential Quantification ("there exists") requires one to be able to ''construct'' the real number ''c'' (in the sense that it can be approximated to any desired precision by a Rational Number ). But if ''f'' hovers near zero during a stretch along its domain, then this cannot necessarily be done. However, constructive analysis provides several alternative formulations of IVT, all of which are equivalent to the usual form in classical analysis, but not in constructive analysis. For example, under the same conditions on ''f'' as in the classical theorem, given any Natural Number ''n'' (no matter how large), there exists (that is, we can construct) a real number ''c''''n'' in the interval such that the Absolute Value of ''f''(''c''''n'') is less than 1/''n''. That is, we can get as close to zero as we like, even if we can't construct a ''c'' that gives us ''exactly'' zero. Alternatively, we can keep the same conclusion as in the classical IVT -- a single ''c'' such that ''f''(''c'') is exactly zero -- while strengthening the conditions on ''f''. |
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