I know that there are things that one can prove using classic approch and not Intuitionistic approch (Using excluded middle will help you with a lot of proves). But are there any examples that works the opposite way ?

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    $\begingroup$ Any intuitionistic proof is also a classic proof. $\endgroup$ – Hagen von Eitzen Jan 6 '17 at 21:00
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    $\begingroup$ I'm not sure what it means, but I recall reading something like "every real-valued function defined on $[0,1]$ is continuous" is an intuitionistic theorem. $\endgroup$ – bof Jan 6 '17 at 21:02
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    $\begingroup$ See Intuitionistic mathematics : The continuum : "In intuitionism, the continuum is both an extension and a restriction of its classical counterpart. In its full form, both notions are incomparable since the intuitionistic real numbers possess properties that the classical real numbers do not have. A famous example is the theorem that in intuitionism every total function on the continuum is continuous." $\endgroup$ – Mauro ALLEGRANZA Jan 6 '17 at 22:02
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    $\begingroup$ "The acceptance of the notion of choice sequence justifies the use of the continuity axioms, from which classically invalid statements can be derived. The weakest of these axioms is the weak continuity axiom. [...] The following theorem is another example of the way in which the continuity axiom refutes certain classical principles : Theorem Every total real function is continuous." $\endgroup$ – Mauro ALLEGRANZA Jan 6 '17 at 22:06

There are many systems of constructive mathematics. Some of them are compatible with classical logic in that all the theorems provable in these systems are classically true. Bishop's system has this property, for example.

Some systems are able to prove theorems that are not classically true. Some common examples of such theorems are:

  • Every function from the real line to the real line is continuous.

  • Every function from the real line to the natural numbers is constant.

  • Every function from the natural numbers to the natural numbers is computable.

These arise as theorems and/or axioms in some systems, but such systems must necessarily have axioms or inference rules that are not classically valid.

There is another, intermediate option. In some other systems, it is not possible to prove any of the three bullets above, but any function that can be explicitly constructed in the system and proved to be a function will have the property.

One example is several systems of higher order Heyting arithmetic. For example, these systems do not prove that every function on the naturals is computable, but each function on the naturals that can be constructed and verified as a function in the system can be shown to be computable by a proof theory argument outside the system.


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