Premium
Finitely Generated Groups and First‐Order Logic
Author(s) -
Morozov A.,
Nies A.
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006106
Subject(s) - stallings theorem about ends of groups , mathematics , word problem (mathematics education) , finitely generated abelian group , turing , finitely generated group , group (periodic table) , order (exchange) , word (group theory) , set (abstract data type) , sentence , combinatorics , discrete mathematics , pure mathematics , arithmetic , computer science , artificial intelligence , chemistry , geometry , organic chemistry , finance , economics , programming language
It is proved that the following classes of finitely generated groups haveΠ 1 1 ‐complete first‐order theories: all finitely generated groups, the n ‐generated groups, and the strictly n ‐generated groups ( n ⩾2). Moreover, all those theories are distinct. Similar techniques show that quasi‐finitely axiomatizable groups have a hyperarithmetical word problem, where a finitely generated group is quasi‐finitely axiomatizable if it is the only finitely generated group satisfying an appropriate first‐order sentence. The Turing degrees of word problems of quasi‐finitely axiomatizable groups form a cofinal set in the Turing degrees of hyperarithmetical sets.