The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist.
For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7].
In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].