In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
Introduction 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.Troelstra, A. S., van Dalen D., Constructivism in mathematics: an introduction 1; Studies in Logic and the Foundations of Mathematics; Springer, 1988; 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.
Logical preliminaries 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.
While a constructive theory proves fewer theorems than its classical counter-part in its classical presentation, it may exhibit attractive meta-logical properties. For example, if a theory exhibits the disjunction property, then if it proves a disjunction then also or . Already in classical arithmetic, this is violated for the most basic propositions about sequences of numbers - as demonstrated next.
Undecidable predicates A common strategy of formalization of real numbers is in terms of sequences of rationals, and so we draw motivation and examples in terms of those. So to define terms, consider a decidable predicate on the naturals, which in the constructive vernacular means is provable, and let be the characteristic function defined to equal exactly where is true. The associated sequence is monotone, with values non-strictly growing between the bounds and . Here, for the sake of demonstration, defining an extensional equality to the zero sequence , it follows that . Note that the symbol "" is used in several contexts here. For any theory capturing arithmetic, there are many yet undecided and even provenly independent such statements . Two -examples are the Goldbach conjecture and the Rosser sentence of a theory.
Consider any theory with quantifiers ranging over primitive recursive, rational-valued sequences. Already minimal logic proves the non-contradiction claim for any proposition, and that the negation of excluded middle for any given proposition would be absurd. 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. No sequence with that disjunction being rejected can be exhibited. Assume the theories at hand are consistent and arithmetically sound. Now Gödel's theorems mean that there is an explicit sequence such that, for any fixed precision, proves the zero-sequence to be a good approximation to , but it can also meta-logically be established that as well as . Here this proposition again amounts to the proposition of universally quantified form. Trivially
even if these disjunction claims here do not carry any information. 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 "" in formalizations below as well. For implications of disjunctions of yet not proven or disproven propositions, one speaks of weak Brouwerian counterexamples.
Order vs. disjunctions The theory of the real closed field may be axiomatized such that all the non-logical axioms are in accordance with constructive principles. This concerns a commutative ring with postulates for a positivity predicate , with a positive unit and non-positive zero, i.e., and . In any such ring, one may define , which constitutes a strict total order in its constructive formulation (also called linear order or, to be explicit about the context, a pseudo-order). As is usual, is defined as .
This first-order theory is relevant as the structures discussed below are model thereof.Erik Palmgren, An Intuitionistic Axiomatisation of Real Closed Fields, Mathematical Logic Quarterly, Volume 48, Issue 2, Pages: 163-320, February 2002 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. This includes "", which will be rendered equivalent to a negation. Crucial disjunctions are now discussed explicitly.
Trichotomy In intuitionistic logic, the disjunctive syllogism in the form generally really only goes in the -direction. In a pseudo-order, one has
and indeed at most one of the three can hold at once. 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 . Other disjunctions are however implied based on other positivity results, e.g. . Likewise, the asymmetric order in the theory ought to fulfill the weak linearity property for all , related to locatedness of the reals.
The theory shall validate further axioms concerning the relation between the positivity predicate and the algebraic operations including multiplicative inversion, as well as the intermediate value theorem for polynomials. In this theory, between any two separated numbers, other numbers exist.
Apartness In the context of analysis, the auxiliary logically positive predicate
may be independently defined and constitutes an apartness relation. With it, the substitute of the principles above give tightness
Thus, apartness can also function as a definition of "", rendering it a negation. All negations are stable in intuitionistic logic, and therefore
The elusive trichotomy disjunction itself then reads
Importantly, a proof of the disjunction carries positive information, in both senses of the word. Via it also follows that . In words: A demonstration that a number is somehow apart from zero is also a demonstration that this number is non-zero. But constructively it does not follow that the doubly negative statement would imply . Consequently, many classically equivalent statements bifurcate into distinct statement. For example, for a fixed polynomial and fixed , the statement that the 'th coefficient of is apart from zero is stronger than the mere statement that it is non-zero. A demonstration of former explicates how 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. In turn, there is then also a strong and a looser notion of, e.g., being a third-order polynomial.
So the excluded middle for is apriori stronger than that for . However, see the discussion of possible further axiomatic principles regarding the strength of "" below.
Non-strict partial order Lastly, the relation may be defined by or proven equivalent to the logically negative statement , and then is defined as . Decidability of positivity may thus be expressed as , which as noted will not be provable in general. But neither will the totality disjunction , see also analytical .
By a valid De Morgan's law, the conjunction of such statements is also rendered a negation of apartness, and so
The disjunction implies , but the other direction is also not provable in general. In a constructive real closed field, the relation "" is a negation and is not equivalent to the disjunction in general.
Variations Demanding good order properties as above but strong completeness properties at the same time implies . Notably, the MacNeille completion has better completeness properties as a collection, but a more intricate theory of its order-relation and, in turn, worse locatedness properties. While less commonly employed, also this construction simplifies to the classical real numbers when assuming .
Invertibility In the commutative ring of real numbers, a provably non-invertible element equals zero. This and the most basic locality structure is abstracted in the theory of Heyting fields.
Formalization Rational sequences A common approach is to identify the real numbers with non-volatile sequences in . The constant sequences correspond to rational numbers. Algebraic operations such as addition and multiplication can be defined component-wise, together with a systematic reindexing for speedup. The definition in terms of sequences furthermore enables the definition of a strict order "" fulfilling the desired axioms. Other relations discussed above may then be defined in terms of it. In particular, any number apart from , i.e. , eventually has an index beyond which all its elements are invertible.Bridges D., Ishihara H., Rathjen M., Schwichtenberg H. (Editors), Handbook of Constructive Mathematics; Studies in Logic and the Foundations of Mathematics; (2023) pp. 201-207 Various implications between the relations, as well as between sequences with various properties, may then be proven.
Moduli As the maximum on a finite set of rationals is decidable, an absolute value map on the reals may be defined and Cauchy convergence and limits of sequences of reals can be defined as usual.
A modulus of convergence is often employed in the constructive study of Cauchy sequences of reals, meaning the association of any to an appropriate index (beyond which the sequences are closer than ) is required in the form of an explicit, strictly increasing function . Such a modulus may be considered for a sequence of reals, but it may also be considered for all the reals themselves, in which case one is really dealing with a sequence of pairs.
Bounds and suprema Given such a model then enables the definition of more set theoretic notions. For any subset of reals, one may speak of an upper bound , negatively characterized using . One may speak of least upper bounds with respect to "". A supremum is an upper bound given through a sequence of reals, positively characterized using "". If a subset with an upper bound is well-behaved with respect to "" (discussed below), it has a supremum.
Bishop's formalization