Linear orderings and powers of characterizable cardinals

We say that a countable model MM completely characterizes an infinite cardinal κκ, if the Scott sentence of MM has a model in cardinality κκ, but no models in cardinality κ+κ+. If a structure MM completely characterizes κκ, κκ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κκ is characterizable if and only if κ=ℵακ=ℵα, for some α<ω1α<ω1 (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κκ is characterizable, then κ+κ+ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κκ is homogeneously   characterizable (cf. Definition 2.2), then κκ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2λκ>2λ is a characterizable cardinal, λλ is characterizable by a densely ordered structure and λλ is the least cardinal such that κλ>κκλ>κ, then κλκλ is also characterizable (Theorem 5.4) and (2) if ℵαℵα and κℵακℵα are characterizable cardinals, then the same is true for κℵα+βκℵα+β, for all countable ββ (Theorem 5.5).Combining these two theorems we get that if κ>2ℵακ>2ℵα is a characterizable cardinal, ℵαℵα is characterizable by a densely ordered structure and ℵαℵα is the least cardinal such that κℵα>κκℵα>κ, then for all β<α+ω1β<α+ω1, κℵβκℵβ is characterizable (Theorem 5.7). Also if κκ is a characterizable cardinal, then κℵακℵα is characterizable, for all countable αα (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].

► We use linearly ordered countable structures to characterize uncountable cardinals. ► The GCH case is known. Our results concern the case where GCH fails. ► The main theorems deal with powers of the form κλκλ, with κκ, λλ characterizable. ► If ℵα,κℵαℵα,κℵα are characterizable, then the same is true for κℵα+βκℵα+β, ββ countable.