Given a model $$\mathcal {M}$$ M of set theory, and a nontrivial automorphism j of $$\mathcal {M}$$ M , let $$\mathcal {I}_{\mathrm {fix}}(j)$$ I fix ( j ) be the submodel of $$\mathcal {M}$$ M whose universe consists of elements m of $$\mathcal {M}$$ M such that $$j(x)=x$$ j ( x ) = x for every x in the transitive closure of m (where the transitive closure of m is computed within $$\mathcal {M}$$ M ). Here we study the class $$\mathcal {C}$$ C of structures of the form $$\mathcal {I}_{\mathrm {fix}}(j)$$ I fix ( j ) , where the ambient model $$\mathcal {M}$$ M satisfies a frugal yet robust fragment of $$\mathrm {ZFC}$$ ZFC known as $$\mathrm {MOST}$$ MOST , and $$j(m)=m$$ j ( m ) = m whenever m is a finite ordinal in the sense of $$\mathcal {M}.$$ M . Our main achievement is the calculation of the theory of $$\mathcal {C}$$ C as precisely $$\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}$$ MOST + Δ 0 P - $$\mathrm {Collection}$$ Collection . The following theorems encapsulate our principal results:
Theorem A.Every structure in $$\mathcal {C}$$ C satisfies $$\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}$$ MOST + Δ 0 P - $$\mathrm { Collection}$$ Collection .
Theorem B. Each of the following three conditions is sufficient for a countable structure $$\mathcal {N}$$ N to be in $$\mathcal {C}$$ C :
(a) $$\mathcal {N}$$ N is a transitive model of $$\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}$$ MOST + Δ 0 P - $$\mathrm {Collection}$$ Collection .
(b) $$\mathcal {N}$$ N is a recursively saturated model of $$\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}$$ MOST + Δ 0 P - $$\mathrm {Collection}$$ Collection .
(c) $$\mathcal {N}$$ N is a model of $$\mathrm {ZFC}$$ ZFC .
Theorem C. Suppose $$\mathcal {M}$$ M is a countable recursively saturated model of $$\mathrm {ZFC}$$ ZFC and I is a proper initial segment of $$\mathrm {Ord}^{\mathcal {M}}$$ Ord M that is closed under exponentiation and contains $$\omega ^\mathcal {M}$$ ω M . There is a group embedding $$j\longmapsto \check{j}$$ j ⟼ j ˇ from $$\mathrm {Aut}(\mathbb {Q})$$ Aut ( Q ) into $$\mathrm {Aut}(\mathcal {M})$$ Aut ( M ) such that Iis the longest initial segment of $$\mathrm {Ord}^{\mathcal {M}}$$ Ord M that is pointwise fixed by $$\check{j}$$ j ˇ for every nontrivial $$j\in \mathrm {Aut}(\mathbb {Q}).$$ j ∈ Aut ( Q ) .
In Theorem C, $$\mathrm {Aut}(X)$$ Aut ( X ) is the group of automorphisms of the structure X, and $$\mathbb {Q}$$ Q is the ordered set of rationals.