[1] 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.
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.
Much of the intricacies of constructive analysis can be framed in terms of the weakness of propositions of the logically negative form
So to define terms, consider a decidable predicate on the naturals, which in the constructive vernacular means
This also means there is no consistent theory (even if anti-classical) rejecting the excluded middle disjunction for any given proposition.
Indeed, it holds that This theorem is logically equivalent to the non-existence claim of a sequence for which the excluded middle disjunction about equality-to-zero would be disprovable.
In the absence of further axioms breaking the meta-logical properties, constructive entailment instead generally reflects provability.
Taboo statements that ought not be decidable (if the aim is to respect the provability interpretation of constructive claims) can be designed for definitions of a custom equivalence "
For implications of disjunctions of yet not proven or disproven propositions, one speaks of weak Brouwerian counterexamples.
The theory of the real closed field may be axiomatized such that all the non-logical axioms are in accordance with constructive principles.
[3] However, this section thus does not concern aspects akin to topology and relevant arithmetic substructures are not definable therein.
As explained, various predicates will fail to be decidable in a constructive formulation, such as these formed from order-theoretical relations.
But the stronger, logically positive law of trichotomy disjunction does not hold in general, i.e. it is not provable that for all reals, See analytical
and the algebraic operations including multiplicative inversion, as well as the intermediate value theorem for polynomials.
In the context of analysis, the auxiliary logically positive predicate may be independently defined and constitutes an apartness relation.
and zero are related, with respect to the ordering predicate on the reals, while a demonstration of the latter shows how negation of such conditions would imply to a contradiction.
By a valid De Morgan's law, the conjunction of such statements is also rendered a negation of apartness, and so The disjunction
Algebraic operations such as addition and multiplication can be defined component-wise, together with a systematic reindexing for speedup.
A modulus of convergence is often employed in the constructive study of Cauchy sequences of reals, meaning the association of any
One formalization of constructive analysis, modeling the order properties described above, proves theorems for sequences of rationals
While this example, an explicit sequence of sums, is a total recursive function to begin with, the encoding also means these objects are in scope of the quantifiers in second-order arithmetic.
In that language, regular rational sequences are degraded to a mere representative of a Cauchy real.
[7] In this context it may also be possible to model a theory or real numbers in terms of Dedekind cuts of
[8] One variant of the diagonal construction relevant for the present context may be formulated as follows, proven using countable choice and for reals as sequences of rationals:[9] Formulations of the reals aided by explicit moduli permit separate treatments.
The Brouwerian school reasons in terms of spreads and adopts the classically valid bar induction.
For example, equality-to-zero is rejected to be decidable when adopting Brouwerian continuity principles or Church's thesis in recursive mathematics.
This article discusses principles compatible with the classical theory and choice is made explicit.
Generally speaking, theorem formulation in constructive analysis mirrors the classical theory closest in separable spaces.
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.
However, as with the intermediate value theorem, an alternative version survives; in constructive analysis, any located subset of the real line has a supremum.