Ever since Cohen invented forcing in 1963, people have studied the properties that cardinals can have in generic extensions of the ground model. A very early result of Lévy shows that if κ is a regular cardinal and λ > κ is strongly inaccessible, then there is a notion of forcing which collapses every cardinal strictly between κ and λ yet preserves every other cardinal. This, of course, answers one question of the genre “What properties can a cardinal have in a generic extension?”

Another question of the same genre that can be asked is the following: Is it possible to have a generic extension of the ground model in which all cardinals are preserved and yet the cofinalities of some cardinals are different? This question was first answered in the affirmative by Prikry, who proved the following theorem.

Theorem 1.1 (Prikry [5]). Assume that V ⊨ “ZFC + κ is measurable”. Then there is a notion of forcing, P, such that for G V-generic over P:

(1) V and V[G] have the same cardinals.

(2) V and V[G] have the same bounded subsets of κ.

(3) V[G], i.e, V[G] ⊨ “κ is Rowbottom”.

(4) V[G] ⊨ “cof(κ) = ω”.

Prikry's result naturally raises the following question: Is it possible to get a generic extension in which cardinals are preserved and yet the cofinalities of certain cardinals are different from the ground model's but are uncountable? This question was first answered in the affirmative by Magidor, who proved the following theorem.

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $-formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings.

The problem of coding relations by means of a single binary relation is well known in the mathematical literature. It was considered in interpretation theory, and also in connection with investigations of decidability of elementary theories. Using various constructions (see, e.g., [2,6], proofs of Theorem 11 in [7] and Theorem 16.51 in [3]), for any model for a countable language, one can construct a model for ℒp (a language with a single binary relation symbol ) in which is interpretable. Each of the mentioned constructions has the same weak point: the universe of is different than the universe of . In [4] we have shown that, in the infinite case, one can eliminate this defect, i.e., for any infinite , we have constructed a model , having the same universe as , in which is elementarily definable. In all constructions mentioned above, it appears that formulas, which define in ( in ), are very complicated. In the present paper, another construction of a model for ℒp is given.

Several results in the theory of recursion in higher types indicate that the effect of a higher type functional on the lower types does not reflect the high type, i.e. the same effect could be obtained by functionals of relatively low type. The two main results here are:

Plus -1 - Theorem (G. Sacks [6] for k = 1, [7] for k > 1). Let H be a normal functional of type ≥ k + 1. Then there exists a normal functional F of type k + 1 such that k-sc(F) = k-sc(H), i.e. the same subsets of tp(k − 1) are recursive in F and H.

Plus - 2 - Theorem (L. Harrington [1]). Let H be a normal functional of type ≥ k + 2. Then there exists a normal functional F of type k +2 such that k-en(H) = k-en(F), i.e. the same subsets of tp(k − 1) are semirecursive in F and H.

The results in this paper also indicate that higher types cannot have too much influence on lower types. The key is the Skolem-Löwenheim theorem. Among the results we mention:

(1) Let n < m. A ⊆ tp(n) × tp(m) be Kleene-semicomputable. Let x ∈ B ⇔ ∀y∈tp(m), ⟨x, y⟩ ∈ A. Then B is . This result may be relativized to a functional of type n + 1.

(2) Let k0 be the type-k-functional that is constant zero. Let F be a functional of type < k. Then, for i ≤ k −2, i-sc(F ,k0) = i-sc(F), i-en(F, k0) = ∀tp(i)(i-en(F)).

By a variety we mean a class of algebras in a language , containing only function symbols, which is closed under homomorphisms, submodels, and products. A variety is said to be strongly abelian if for any term in , the quasi-identity

holds in .

In [1] it was proved that if a strongly abelian variety has less than the maximal possible uncountable spectrum, then it is equivalent to a multisorted unary variety. Using Shelah's Main Gap theorem one can conclude that if is a classifiable (superstable without DOP or OTOP and shallow) strongly abelian variety then is a multisorted unary variety. In fact, it was known that this conclusion followed from the assumption of superstable without DOP alone.

This paper is devoted to the proof that the superstability assumption is enough to obtain the same structure result. This fulfills a promise made in [2]. Namely, we will prove the following

Theorem 0.1. If is a superstable strongly abelian variety, then it is multisorted unary.