{"id":144,"date":"2019-02-01T21:20:07","date_gmt":"2019-02-01T21:20:07","guid":{"rendered":"http:\/\/jkhorner.com\/?page_id=144"},"modified":"2021-01-10T19:29:32","modified_gmt":"2021-01-10T19:29:32","slug":"geometry","status":"publish","type":"page","link":"https:\/\/jkhorner.com\/index.php\/geometry\/","title":{"rendered":"GEOMETRY"},"content":{"rendered":"<p>Tarski\u2019s \u201cweak-continuity\u201d elementary geometry (TWCEG) is a first-order finite axiomatization of Euclidean geometry. TWCEG is consistent and incomplete. Its universal sentences are decidable. There is no known proof of the consistency or independence of TWCEG that uses a finite-domain model. <a href=\"\/GEOMETRY\/JKH20201228_0734CT_CCWCTEG.pdf\">Here<\/a> I show that if we replace the \u201csegment construction\u201d axiom of TWCEG with a slightly weaker axiom, we obtain an elementary Euclidean geometry (which I call \u201cCCWCEG\u201d) that is first-order, finitely axiomatized, incomplete, consistent, and independent. The proofs of the consistency and independence of CCWCEG require only finite-domain models. I further show that the universal sentences of CCWCEG are decidable.<\/p>\n<p>The manuscript has two high-level parts \u2013 a ~15 pp. body, which can be read standalone, and ~150 pp. of technical appendices, which contain the detailed proof of the independence of CCWCEG.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tarski\u2019s \u201cweak-continuity\u201d elementary geometry (TWCEG) is a first-order finite axiomatization of Euclidean geometry. TWCEG is consistent and incomplete. Its universal sentences are decidable. There is no known proof of the consistency or independence of TWCEG that uses a finite-domain model. Here I show that if we replace the \u201csegment construction\u201d axiom of TWCEG with a &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/jkhorner.com\/index.php\/geometry\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;GEOMETRY&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-144","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/comments?post=144"}],"version-history":[{"count":17,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/144\/revisions"}],"predecessor-version":[{"id":359,"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/pages\/144\/revisions\/359"}],"wp:attachment":[{"href":"https:\/\/jkhorner.com\/index.php\/wp-json\/wp\/v2\/media?parent=144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}