On expandability of models of Peano arithmetic. III (Q1238808)

[For parts I, II see ibid. 35, 409-419 and 421-431 (1976; Zbl 0353.02035 and 353.02036).] The paper is devoted to two recent results on expandability of models for Peano arithmetic. The first one, due to G. Wilmers, says that if \(\mathfrak M\models P^{ZF}\) is countable and nonstandard then the following conditions are equivalent: 1. \(\mathfrak M\) is \(ZF\)-expandable, 2. \(\mathfrak M\) is \(A_2\)-expandable, 3. \(\mathfrak M\) is \(A_2^-\)-expandable, 4. for every \(n\geq1,\mathfrak M\) is \(A_2^-|\Delta_n^1\), expandable, 5. for some \(n\geq1\), \(\mathfrak M\) is \(A_2^-|\Delta_n^1\)-expandable, 6. \(\mathfrak M\) is recursively saturated, 7. there is a Scott set \(A\) such that \(\mathfrak M\) is \(A\)-saturated and \(Th(\mathfrak M)\in A\). (\(A\) set \(A\subseteq\mathcal P(\omega)\) is said to be a Scott set iff \(A\) is recursively closed and satisfies König's lemma. A structure \(\mathfrak A\) is said to be \(A\)-saturated for a given \(A\subseteq\mathcal P(\omega)\) iff \(\mathfrak A\) realizes every partial type \(\Gamma(x)\) with parameters from \(|\mathfrak A|\) such that \(\Gamma\) is consistent with \(Th(\mathfrak A)\) and \(\Gamma\in A\) and \(\mathfrak A\) omits any partial type \(\Gamma\) which is not in \(A\).) The second result of the paper is devoted to assertions about arithmetical analogons of \(Z\). Ratajczyk' s results onaxiomatizability of certain extensions of \(ZF\). Namely it can be proved that \(P^{A_2^-}\); has the following axiomatics: \(P\cup\{\varphi\to\operatorname{Con}(S\cup\{\varphi\}):\varphi\in L(P), S\subseteq A_2^-,S\text{ finite}\}\). Using the ideas of ramified analysis one can show also that the theory \(P^{A_2^-|\Sigma_n^1}(n\geq 1)\) can be axiomatized by: \(P \cup\{\varphi\to\operatorname{Con}(S\cup\{\varphi\}):\varphi\in L(P), \varphi\) is a sentence, \(S\subseteq T_n,S\) is finite\}, where \(T_n\) is the theory whose non-logical axioms are the set existence schemae for hereditary \(\Sigma_n^1\) formulas without set parameters and basis theorems for this same class of formulas.