Let κ and λ be infinite cardinals such that λ ≤ λ (we have new information for the case when κ ≤ λ). Let T be a theory in Lκ +, ω of cardinality at most κ, let . Now define
Our main concept in this paper is is a theory in Lκ +, ω of cardinality κ at most, and φ(x, y) ϵ Lκ +, ω}. This concept is interesting because of
Theorem 1. Let T ⊆ Lκ +, ω of cardinality ≤ κ, and. If
then (∀χ > κ)I(χ, T) = 2χ (where I(χ, T) stands for the number of isomorphism types of models of T of cardinality χ).
Many years ago the second author proved that . Here we continue that work by proving
Theorem 2. .
Theorem 3. For everyκ ≤ λwe have.
For some κ or λ we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem.
Theorem 4. For every T ⊆ Lκ +, ω, and any set of formulas ⊆ Lκ +, ω such thatT ⊇ Lκ +, ω, if T is (, μ)-unstable for μ satisfyingμμ*(λ,κ) = μ then T is-unstable (i.e. for every χ ≥ λ, T is (, χ)-unstable). Moreover, T is Lκ +, ω-unstable.
In the second part of the paper, we show that always in the applications it is possible to replace the function I(χ, T) by the function IE(χ, T), and we give an application of the theorems to Boolean powers.