In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. The name of the subject contrasts with classical analysis, which in this context means analysis done according to the more common principles of classical mathematics. However, there are various schools of thought and many different formalizations of constructive analysis. Whether classical or constructive in some fashion, any such framework of analysis axiomatizes the real number line by some means, a collection extending the rationals and with an apartness relation definable from an asymmetric order structure. Center stage takes a positivity predicate, here denoted , which governs an equality-to-zero . The members of the collection are generally just called the real numbers. While this term is thus overloaded in the subject, all the frameworks share a broad common core of results that are also theorems of classical analysis. Constructive frameworks for its formulation are extensions of Heyting arithmetic by types including , constructive second-order arithmetic, or strong enough topos-, type- or constructive set theories such as , a constructive counter-part of . Of course, a direct axiomatization may be studied as well. The base logic of constructive analysis is intuitionistic logic, which means that the principle of excluded middle is not automatically assumed for every proposition. If a proposition is provable, this exactly means that the non-existence claim being provable would be absurd, and so the latter cannot also be provable in a consistent theory. The double-negated existence claim is a logically negative statement and implied by, but generally not equivalent to the existence claim itself. Much of the intricacies of constructive analysis can be framed in terms of the weakness of propositions of the logically negative form , which is generally weaker than . In turn, also an implication can generally be not reversed.

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Constructive set theory
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories. In addition to rejecting the principle of excluded middle (), constructive set theories often require some logical quantifiers in their axioms to be set bounded, motivated by results tied to impredicativity.
Heyting arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Heyting arithmetic can be characterized just like the first-order theory of Peano arithmetic , except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle , do not hold.
Markov's principle
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function.
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