Canonical reconstruction and forcing absoluteness of standard structures

Abstract

We isolate a simple preservation principle governing when it is absolute, between transitive models of set theory, that a given algebraic or topological-algebraic structure has a standard form F(X) indexed by a set X. The principle is: if the index X (or a proxy for it) can be recovered from F(X) by a uniform definable construction, then the class of structures isomorphic to some F(X) is downward absolute from forcing extensions. Answering a question raised by Noah Schweber, we deduce in particular that no group that fails to be a full symmetric group in the ground model can become one after forcing; the result holds already in \(\textsf{ZF}\). The same mechanism applies to full transformation monoids, powerset Boolean algebras, full relation algebras, full clones, full partition lattices, products \(R^X\) of finitely generated centrally indecomposable rings, the commutative \(C^*\)-algebras \(\ell _\infty (X)\) and \(c_0(X)\), full endomorphism rings, the operator algebras \(\mathcal {B}(H)\) and \(\mathcal {K}(H)\), and \(\ell _1(X)\) as a real Banach lattice. In the motivating symmetric-group case, the same reconstruction gives more than descent: it yields a uniform \(\Pi ^1_1\) definition of fullness over transitive \(\textsf{ZF}\)-models. We then exhibit clean torsor obstructions, in the standard symmetric-model situation: finite covers \(Y \times n\) already separate \(\textsf{ZF}\)-failure from \(\textsf{ZFC}\)-descent without any completeness caveat, and the finite-support normed space \(c_{00}(I)\) provides the analogous Banach example. Bare-Banach-space isomorphism with \(\ell _1(\Gamma )\) exhibits a genuine \(\textsf{ZFC}\)-descent. We conclude with the corresponding, relative, obstructions to \(\Pi ^1_1\)-definability of standardness over transitive \(\textsf{ZF}\)-models.