Effective Extensions of Linear Forms on a Recursive Vector Space Over a Recursive Field

✍ Scribed by R. G. Downey; Iraj Kalantari

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 491 KB
• Volume: 31
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19850311302

✦ Synopsis

51. Introduction

In recent years. many authors have become reinterested in uhat is called the effective co?ste?it of various mat>hematical systems. That is. authors tend t'o ask questions such as "if we give a structure certain effectivity (identified here as "recursive") conditions \hat other effectivity conditions a,re guaranteed ". The pioneering paper in the modern approach to such questions was that. of METAKIDES and NERODE [ll] where t.he authors invcstigated t.he lattice L( V,) of recursively enumerable (r.e.) subspaces of an infinite dimensional. fully effective, vector space V , . There, V , is a recursive set. whose basic opcrabions of addition, scalar multiplication. and determining independence are recursive functions and whose underlying field is similarly recursive. Amongst ot,lier results. they show the usual process of extending an independent set is not intrinsically effective even in a structure such as V , . Namely. they show that there exist's a recursive independent subset I of 8, such that v& modulo the span of I is infinite dimensional and such that if J is an r.e. hdependent superset of I . then J -I is finite. Also they use V , to show that thei: recursively presented vector spaces without any recursive bases.