2012 seminar talk: Borel structures and Borel theories
Talk held by André Nies (University of Auckland, New Zealand) at the KGRC seminar on 2012-01-19.
Continuum size structures occur naturally in analysis, algebra, and other areas. Examples are the additive group of real numbers, and the ring of continuous functions on the unit interval. How about effectiveness constraints on their presentations? A reasonable approach is to require that domain and relations are Borel. Structures from these areas usually have such Borel presentations.
Borel structures were introduced by H. Friedman in 1978. He proved among other things that each countable theory has a Borel model of size the continuum. Hjorth and I considered the case where the language is uncountable but Borel (for instance, the language of a vector space over the reals). In a JSL paper (June 2011) we show that the completeness theorem fails for Borel structures of this kind: some complete Borel theory has no Borel model.
I will also include open questions. For instance, does every Borel field Borel embed into a Borel algebraically closed field? If not, this would yield an alternative proof of my result with Hjorth. A further long-standing question is whether the Boolean algebra of subsets of natural numbers modulo finite sets has an injective Borel presentation.