To be is to be finitely supported: symmetry, definability, and mathematical existence

Abstract The 1905 exchange between Hadamard, Lebesgue, Borel, and Baire exposed a lasting tension in the foundations of mathematics: must a mathematical object be definable in order to exist, and if so, in what sense of “definable”? In this sense, the paper proposes symmetric definability as a criterion of mathematical admissibility: an object is admitted not merely when it is syntactically describable, constructively generated, or set-theoretically coded, but when its identity is determined by all admissible symmetries once finitely many relevant parameters have been fixed. This criterion is implemented in the theory of finitely supported sets, in the Fraenkel–Mostowski hierarchy over an infinite set of atoms $$\mathrm{A}$$ with an admissible permutation group $$G \leq \mathrm{Sym}(\mathrm{A})$$ . The paper has four technical components. First, the universe $$\mathrm{FS}_{G}{(\mathrm{A})}$$ of hereditarily finitely supported sets is introduced along with the core support lemmas. Second, a support–definability correspondence is proved for finite-arity relations on atoms in two canonical settings: finite support is equivalent to first-order definability from finitely many parameters, with quantifier-free witnesses. Third, a semantic–syntactic gap is identified in the uncountable case, where finite support cannot be captured by definability in any fixed countable supported language. Fourth, three case studies show how the choice of symmetry group $$G$$ determines internal choice, orderability, and cardinal behaviour. The paper also clarifies the scope of the criterion for classical analysis, metric spaces, and tame geometry: pure-set representations are trivially supported, while atom-dependent representations are admitted exactly when their defining data are finitely supported. This work does not propose finitely supported sets as the exclusive foundation for mathematics. Instead, it argues that finite support offers a precise and philosophically illuminating criterion of admissibility within a symmetry-based framework. In this context, mathematical existence is linked not to explicit construction or mere consistency but to invariance under admissible symmetries with finite support.