Math 2.0

Mathematical logic and foundations Definition

Constructive analysis

Builds on ConstructivismReal numbersCauchy sequenceIntermediate value theoremLaw of excluded middle

Definition

Constructive analysis is real and functional analysis developed within constructivism, with intuitionistic logic. In Bishop's approach, the basic notions are these.

  1. A real number is a sequence (xn)n≥1(x_n)_{n \ge 1} of rational numbers that is regular: ∣xm−xn∣≤1m+1n\abs{x_m - x_n} \le \frac{1}{m} + \frac{1}{n} for all m,n≥1m, n \ge 1.
  2. Two real numbers (xn)(x_n) and (yn)(y_n) are equal if ∣xn−yn∣≤2n\abs{x_n - y_n} \le \frac{2}{n} for all n≥1n \ge 1.
  3. A real number (xn)(x_n) is positive if xn>1nx_n > \frac{1}{n} for some nn, and x<yx < y means that y−xy - x is positive.

Functions, continuity and other notions are then defined so that each existence statement comes with a method of construction.

Remarks

A regular sequence is a Cauchy sequence with an explicit modulus of convergence, and it defines a classical real number. Example: the sequence xn=∑k=0n1k!x_n = \sum_{k=0}^{n} \frac{1}{k!} is regular, since ∣xm−xn∣≤∑k>n1k!≤1n\abs{x_m - x_n} \le \sum_{k > n} \frac{1}{k!} \le \frac{1}{n} for m>n≥1m > n \ge 1; it defines ee. Non-example: the statement "for every real xx, either x=0x = 0 or x≠0x \ne 0" is not a theorem of constructive analysis. A proof would give a method to decide, for each regular sequence, whether all its terms stay within 2n\frac{2}{n} of 00. Such a method would need the law of excluded middle for statements about infinitely many terms. The intermediate value theorem holds in an approximate form: if ff is continuous on [0,1][0, 1] with f(0)<0<f(1)f(0) < 0 < f(1), then for each ε>0\varepsilon > 0 there is xx with ∣f(x)∣<ε\abs{f(x)} < \varepsilon.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptLaw of excluded middle, statement, no evidence levelLaw of excluded middlestatementModus ponens, concept, no evidence levelModus ponensconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptIntermediate value theorem, statement, no evidence levelIntermediate value th…statementTautology, definition, no evidence levelTautologydefinitionConstructive proof, concept, no evidence levelConstructive proofconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptConstructivism, definition, no evidence levelConstructivismdefinitionConstructive analysis, definition, no evidence levelConstructive analysisdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Proposition concept
  3. Real numbers object
  4. Sequence definition
  5. Limit of a function definition
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Predicate definition
  11. Real sequence definition
  12. Continuity at a point definition
  13. Biconditional definition
  14. Truth table concept
  15. Law of excluded middle statement
  16. Modus ponens concept
  17. Existential quantifier definition
  18. Cauchy sequence definition
  19. Continuous function definition
  20. Propositional logic concept
  21. Mathematical proof concept
  22. Intermediate value theorem statement
  23. Tautology definition
  24. Constructive proof concept
  25. Logical equivalence definition
  26. Double negation law statement
  27. Proof by contradiction concept
  28. Constructivism definition
  29. Constructive analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Constructive_analysis
About this page
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Kinddefinition
WritingW0 stub
MSC03F60, 03F65
DefinesConstructive analysis
UsesConstructivism, Real numbers, Cauchy sequence, Intermediate value theorem, Law of excluded middle
Changed2 November 2026

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