Finitely generated groups and first-order logic

• Autores: A. Morozov, André Nies
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 71, Nº 3, 2005, págs. 545-562
• Idioma: inglés
• DOI: 10.1112/s0024610704006106
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• Resumen

  • It is proved that the following classes of finitely generated groups have -complete first-order theories: all finitely generated groups, the -generated groups, and the strictly -generated groups (). 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.