Frege’s system in "Grundgesetze der Arithmetik"

E18534
formal logical system foundational system for arithmetic second-order logical system

Frege’s system in "Grundgesetze der Arithmetik" is a foundational logical framework for arithmetic based on second-order logic and Basic Law V, whose inconsistency—revealed by Russell’s paradox—marked a turning point in the development of modern logic and set theory.

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Observed surface forms (4)

Surface form Occurrences
Grundgesetze der Arithmetik 2
Fregean function–argument analysis 1
Fregean quantifier notation 1

Statements (49)

Predicate Object
instanceOf formal logical system ⓘ
foundational system for arithmetic ⓘ
second-order logical system ⓘ
aimsTo provide a logical foundation for arithmetic ⓘ
basedOn second-order logic ⓘ
centralAxiom Basic Law V ⓘ
creator Gottlob Frege ⓘ
describedInWork Frege’s system in "Grundgesetze der Arithmetik" self-linksurface differs ⓘ
surface form: Grundgesetze der Arithmetik
formalizes Hume’s Principle (derivable, not postulated) ⓘ
formalLanguage ideography (Begriffsschrift) ⓘ
foundInVolume Frege’s system in "Grundgesetze der Arithmetik" self-linksurface differs ⓘ
surface form: Grundgesetze der Arithmetik, Volume I

Grundgesetze der Arithmetik, Volume II ⓘ
goal derive Peano axioms for arithmetic from purely logical principles ⓘ
hasComponent axioms for identity ⓘ
axioms for quantification ⓘ
axioms for truth-functions ⓘ
definition of finite cardinal numbers ⓘ
definition of numbers as extensions ⓘ
definition of successor ⓘ
proofs of basic laws of arithmetic ⓘ
historicalImpact influenced development of axiomatic set theory ⓘ
influenced development of type theory ⓘ
influenced later work in model theory and proof theory ⓘ
triggered crisis in foundations of mathematics ⓘ
includesAxiom Basic Law V ⓘ
inconsistencyRevealedBy Russell’s paradox ⓘ
inconsistencySource Basic Law V ⓘ
influenced Alfred North Whitehead ⓘ
Bertrand Russell ⓘ
Principia Mathematica ⓘ
Zermelo–Fraenkel set theory ⓘ
surface form: Zermelo–Fraenkel set theory (indirectly)

neo-logicist programs in the philosophy of mathematics ⓘ
intendedToShow that arithmetic is reducible to logic ⓘ
isInconsistent true ⓘ
logicalFramework axiomatic calculus for functions and objects ⓘ
logicalNotion course-of-values operator (extension operator) ⓘ
paradoxType set-theoretic paradox of the extension of the concept "not self-membered" ⓘ
philosophicalProgram logicism ⓘ
quantificationType second-order quantification over concepts ⓘ
responseByFrege attempted modification of Basic Law V in Appendix to Volume II ⓘ
treats numbers as extensions of concepts ⓘ
treatsAsObjects extensions of concepts ⓘ
uses extensionality for concepts ⓘ
function–argument analysis of propositions ⓘ
truth-values as objects ⓘ
usesDistinction between objects and concepts ⓘ
between sense and reference (in the surrounding theory) ⓘ
yearFirstVolumePublished 1893 ⓘ
yearSecondVolumePublished 1903 ⓘ

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Frege’s system in "Grundgesetze der Arithmetik" → describedInWork → Frege’s system in "Grundgesetze der Arithmetik" self-linksurface differs ⓘ
this entity surface form: Grundgesetze der Arithmetik
Frege’s system in "Grundgesetze der Arithmetik" → foundInVolume → Frege’s system in "Grundgesetze der Arithmetik" self-linksurface differs ⓘ
this entity surface form: Grundgesetze der Arithmetik, Volume I
Begriffsschrift → introducedConcept → Frege’s system in "Grundgesetze der Arithmetik" ⓘ
this entity surface form: Fregean quantifier notation
Begriffsschrift → introducedConcept → Frege’s system in "Grundgesetze der Arithmetik" ⓘ
this entity surface form: Fregean function–argument analysis
Gottlob Frege → notableWork → Frege’s system in "Grundgesetze der Arithmetik" ⓘ
this entity surface form: Grundgesetze der Arithmetik
Russell’s paradox → undermined → Frege’s system in "Grundgesetze der Arithmetik" ⓘ