Matches in SemOpenAlex for { <https://semopenalex.org/work/W4296278112> ?p ?o ?g. }
Showing items 1 to 30 of
30
with 100 items per page.
- W4296278112 abstract "Euclid's reasoning is essentially constructive. Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive, even if we restrict attention (as we do here) to the theory with line-circle and circle-circle continuity in place of first-order Dedekind completeness. Here we exhibit three constructive versions of Tarski's theory. One, like Tarski's theory, has existential axioms and no function symbols. We then consider a version in which function symbols are used instead of existential quantifiers. The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Pasch's axiom and the parallel axiom; this choice of function symbols connects directly to ruler-and-compass constructions. All three versions have this in common: the axioms have been modified so that the points they assert to exist are unique and depend continuously on parameters. This modification of Tarski's axioms, with classical logic, has the same theorems as Tarski's theory, but we obtain results connecting it with ruler-and-compass constructions as well. In particular, points constructively proved to exist can be constructed with ruler and compass, uniformly in parameters; the same is true with non-constructive proofs if several constructions are allowed for different cases." @default.
- W4296278112 created "2022-09-19" @default.
- W4296278112 creator A5034098863 @default.
- W4296278112 date "2014-07-13" @default.
- W4296278112 modified "2023-09-24" @default.
- W4296278112 title "A Constructive Version of Tarski's Geometry" @default.
- W4296278112 doi "https://doi.org/10.48550/arxiv.1407.4399" @default.
- W4296278112 hasPublicationYear "2014" @default.
- W4296278112 type Work @default.
- W4296278112 citedByCount "0" @default.
- W4296278112 crossrefType "posted-content" @default.
- W4296278112 hasAuthorship W4296278112A5034098863 @default.
- W4296278112 hasBestOaLocation W42962781121 @default.
- W4296278112 hasLocation W42962781121 @default.
- W4296278112 hasLocation W42962781122 @default.
- W4296278112 hasOpenAccess W4296278112 @default.
- W4296278112 hasPrimaryLocation W42962781121 @default.
- W4296278112 hasRelatedWork W1542130776 @default.
- W4296278112 hasRelatedWork W1573969001 @default.
- W4296278112 hasRelatedWork W2031104901 @default.
- W4296278112 hasRelatedWork W2081484232 @default.
- W4296278112 hasRelatedWork W21991091 @default.
- W4296278112 hasRelatedWork W2767543552 @default.
- W4296278112 hasRelatedWork W3097276150 @default.
- W4296278112 hasRelatedWork W35847595 @default.
- W4296278112 hasRelatedWork W4240255957 @default.
- W4296278112 hasRelatedWork W2472100930 @default.
- W4296278112 isParatext "false" @default.
- W4296278112 isRetracted "false" @default.
- W4296278112 workType "article" @default.