Grundlagen der Geometrie¶
Hilbert, D. (1899). Grundlagen der Geometrie.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Abstraction
This sourceAxiomatic abstraction of geometry into undefined primitives and axioms independent of intuitive content.
- Completeness
- … proof from the axioms (first-order classical logic is complete with respect to its model-theoretic semantics — Gödel's completeness theorem of 1929 — but Peano arithmetic with respect to its standard model is not complete, because Gödel's incompleteness theorem of 1931 exhibits true-but-unprovable statements);
This sourceTeubner, Leipzig. Develops axiomatic abstraction: geometry is determined entirely by a set of axioms (incidence, betweenness, congruence, continuity) independent of intuitive content, the prerequisite move that made later metalogical completeness questions formulable.
- … proof from the axioms (first-order classical logic is complete with respect to its model-theoretic semantics — Gödel's completeness theorem of 1929 — but Peano arithmetic with respect to its standard model is not complete, because Gödel's incompleteness theorem of 1931 exhibits true-but-unprovable statements);
- Formalization
- The intellectual high-water mark of this move is the early-twentieth-century formalist program in mathematics, where Hilbert (1899/1902) re-grounded Euclidean geometry on a set of explicit, gap-free axioms precisely to expose every assumption that classical practice had carried silently.
This sourceRe-grounds Euclidean geometry on ~20 explicit, gap-free axioms in five groups (incidence, order, congruence, parallels, continuity), exposing assumptions classical practice carried silently and making it possible to ask which theorems depend on which axioms.
Supported in partVerified against the work's full text
Hilbert's text sets out to give a complete, simple axiom system for geometry in five groups and to show the role of each, which supports the axiomatization part of the claim.
“ein Tollständiges und möglichst einfaches System von Axiomen”
- The intellectual high-water mark of this move is the early-twentieth-century formalist program in mathematics, where Hilbert (1899/1902) re-grounded Euclidean geometry on a set of explicit, gap-free axioms precisely to expose every assumption that classical practice had carried silently.
Verification¶
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