Math 2.0

Mathematical logic and foundations Definition

Foundations of mathematics

Builds on Predicate logicMathematical proof

Definition

The foundations of mathematics is the part of mathematical logic and philosophy that studies the basic framework of mathematics. It studies:

  1. the formal languages in which mathematical statements are written, such as predicate logic;
  2. the axiom systems from which mathematical theorems are derived, such as Zermelo–Fraenkel set theory;
  3. the notion of proof, and the properties of the whole framework, such as its consistency and its completeness.

Remarks

For example, the Peano axioms can be interpreted in set theory. Each natural number becomes a set, and each axiom becomes a theorem of set theory. A theorem about prime numbers alone, such as the infinitude of primes, is not a foundational result: it is about objects inside the framework, not about the framework. The main schools of the early twentieth century were logicism, formalism and intuitionism.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptFoundations of mathematics, definition, no evidence levelFoundations of mathem…definition
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. Proposition concept
  2. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Predicate definition
  7. Biconditional definition
  8. Modus ponens concept
  9. Mathematical proof concept
  10. Propositional logic concept
  11. Predicate logic concept
  12. Foundations of mathematics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Foundations_of_mathematics
About this page
TagQ24EQE2HVG
URIm2://Q24EQE2HVG
Kinddefinition
WritingW0 stub
MSC03A05, 03B30
DefinesFoundations of mathematics
UsesPredicate logic, Mathematical proof
Changed2 November 2026

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