Editing Constructive analysis (section)

==Theorems==
Many classical theorems can only be proven in a formulation that is [[logically equivalent]], over [[classical logic]]. Generally speaking, theorem formulation in constructive analysis mirrors the classical theory closest in [[separable space]]s. Some theorems can only be formulated in terms of [[approximation]]s.

===The intermediate value theorem===
For a simple example, consider the [[intermediate value theorem]] (IVT).
In classical analysis, IVT implies that, given any [[continuous function]] ''f'' from a [[closed interval]] [''a'',''b''] to the [[real line]] ''R'', if ''f''(''a'') is [[negative number|negative]] while ''f''(''b'') is [[positive number|positive]], then there exists a [[real number]] ''c'' in the interval such that ''f''(''c'') is exactly [[0 (number)|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''<sub>''n''</sub> in the interval such that the [[absolute value]] of ''f''(''c''<sub>''n''</sub>) 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''.
We require that ''f'' be ''locally non-zero'', meaning that given any point ''x'' in the interval [''a'',''b''] and any natural number ''m'', there exists (we can construct) a real number ''y'' in the interval such that |''y'' - ''x''| < 1/''m'' and |''f''(''y'')| > 0.
In this case, the desired number ''c'' can be constructed.
This is a complicated condition, but there are several other conditions that imply it and that are commonly met; for example, every [[analytic function]] is locally non-zero (assuming that it already satisfies ''f''(''a'') < 0 and ''f''(''b'') > 0).

For another way to view this example, notice that according to [[classical logic]], if the ''locally non-zero'' condition fails, then it must fail at some specific point ''x''; and then ''f''(''x'') will equal 0, so that IVT is valid automatically.
Thus in classical analysis, which uses classical logic, in order to prove the full IVT, it is sufficient to prove the constructive version. From this perspective, the full IVT fails in constructive analysis simply because constructive analysis does not accept classical logic. Conversely, one may argue that the true meaning of IVT, even in classical mathematics, is the constructive version involving the ''locally non-zero'' condition, with the full IVT following by "pure logic" afterwards.
Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach gives a better insight into the true meaning of theorems, in much this way.

===The least-upper-bound principle and compact sets===
Another difference between classical and constructive analysis is that constructive analysis does not prove the [[least-upper-bound principle]], i.e. that any [[subset]] of the real line '''R''' would have a [[least upper bound]] (or supremum), possibly infinite.
However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any ''located'' subset of the real line has a supremum.
(Here a subset ''S'' of '''R''' is ''located'' if, whenever ''x'' < ''y'' are real numbers, either there exists an element ''s'' of ''S'' such that ''x'' < ''s'', [[logical disjunction|or]] ''y'' is an [[upper bound]] of ''S''.)
Again, this is classically equivalent to the full least upper bound principle, since every set is located in classical mathematics.
And again, while the definition of located set is complicated, nevertheless it is satisfied by many commonly studied sets, including all [[Interval (mathematics)|intervals]] and all [[compact set]]s.

Closely related to this, in constructive mathematics, fewer characterisations of [[compact space]]s are constructively valid—or from another point of view, there are several different concepts that are classically equivalent but not constructively equivalent.
Indeed, if the interval [''a'',''b''] were [[sequentially compact]] in constructive analysis, then the classical IVT would follow from the first constructive version in the example; one could find ''c'' as a [[cluster point]] of the [[infinite sequence]] (''c''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub>.