Beth's property fails in L<ω

Abstract

In this paper we prove that Beth's property does not hold in L<ω. This answers a question posed by Magidor and Malitz in [8]. Beth's property is a natural closure condition on a language which says that everything implicitly definable in the language is also explicitly definable in the language. That the first-order predicate calculus (L0) has the property was first shown by Beth [4]. Lopez-Escobar proved that also has Beth's property [7]. Malitz [9] showed the Beth's property fails in Lκλ where κ ≥ λ ≥ ω1. Friedman and Silver showed that the property fails in Lκλ for κ > ω1. Also Friedman [5] showed that extensions of elementary logic using cardinality quantifiers (L)1κ do not have Beth's property.The undefined notation used here is standard. If further clarification is needed, we refer the reader to [3]. κ and λ denote infinite cardinals. cX and ȣX∣ denote the cardinality of X. cfμ will denote the cofinality of the order μ. All languages discussed are assumed to have no function or constant symbols. All structures are relational. The type of a formula is the set of relation symbols appearing in the formula (excluding equality). The type of a set of formulas is the set of all relation symbols appearing in some formula of the set of formulas. For purpose of this paper we will assume that to each n-ary relation in a structure there is associated an n-ary relation symbol and that in any given structure this association is one-to-one. Using this convention we can define the type of a structure to be the set of those relation symbols to which there is associated a relation in the structure. Sometimes we use the term predicate instead of relation.