Math 2.0

Real functions Definition

Nonstandard analysis

Builds on Real numbersHyperreal numberTransfer principleStandard part functionDerivative

Definition

Nonstandard analysis is the part of mathematics that develops analysis with the following tools.

  1. A proper extension ∗ℝ{}^*\R of the real numbers, such as the field of hyperreal numbers, that contains nonzero infinitesimals.
  2. For each set, function and relation on ℝ\R, an extension of it to ∗ℝ{}^*\R.
  3. The transfer principle, under which a first-order statement about ℝ\R holds if and only if the corresponding statement about ∗ℝ{}^*\R holds.

Notions of analysis are then expressed with infinitesimals and the standard part function.

Remarks

Example: in nonstandard analysis, a function f ⁣:ℝ→ℝf \colon \R \to \R has derivative LL at xx if st⁡((∗f(x+ε)−f(x))/ε)=L\operatorname{st}\bigl(({}^*f(x + \varepsilon) - f(x))/\varepsilon\bigr) = L for every nonzero infinitesimal ε\varepsilon, where ∗f{}^*f is the extension of ff to ∗ℝ{}^*\R. For f(x)=x2f(x) = x^2 the quotient is 2x+ε2x + \varepsilon, whose standard part is 2x2x.

Non-example: smooth infinitesimal analysis also uses infinitesimals, but they are nilpotent and its logic is intuitionistic. It does not use an extension of ℝ\R with transfer, so it is not nonstandard analysis.

The theorems of nonstandard analysis about standard objects can also be proved by ordinary methods, but the nonstandard proofs are often shorter.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Derivative, concept, no evidence levelDerivativeconceptHyperreal number, concept, no evidence levelHyperreal numberconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectStandard part function, concept, no evidence levelStandard part functionconceptUltrafilter, concept, no evidence levelUltrafilterconceptUltraproduct, concept, no evidence levelUltraproductconceptWell-formed formula, concept, no evidence levelWell-formed formulaconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionŁoś's theorem, statement, no evidence levelŁoś's theoremstatementBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptTransfer principle, statement, no evidence levelTransfer principlestatementNonstandard analysis, definition, no evidence levelNonstandard 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. Derivative concept
  2. Hyperreal number concept
  3. Proposition concept
  4. Real numbers object
  5. Standard part function concept
  6. Ultrafilter concept
  7. Ultraproduct concept
  8. Well-formed formula concept
  9. Conjunction definition
  10. Disjunction definition
  11. Implication definition
  12. Negation definition
  13. Predicate definition
  14. Łoś's theorem statement
  15. Biconditional definition
  16. Propositional logic concept
  17. Predicate logic concept
  18. Transfer principle statement
  19. Nonstandard analysis definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Nonstandard_analysis
About this page
TagG00EDC2ET4
URIm2://G00EDC2ET4
Kinddefinition
WritingW0 stub
MSC26E35, 03H05
DefinesNonstandard analysis
UsesReal numbers, Hyperreal number, Transfer principle, Standard part function, Derivative
Changed2 November 2026

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