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The true theory of second-order arithmetic consists of all the sentences in the language of second-order arithmetic that are satisfied by the standard model of second-order arithmetic, whose first-order part is the structure and whose second-order part consists of every subset of .

The true theory of first-order arithmetic, 1=Th($\backslash mathcal\{N/math>)\}\},\; is\; a\; subset\; of\; the\; true\; theory\; of\; second-order\; arithmetic,\; and\; 1=Th($ \backslash mathcal\{N/math>)\}\}\; is\; definable\; in\; second-order\; arithmetic.\; However,\; the\; generalization\; of\; Post\text{'}s\; theorem\; to\; theanalytical\; hierarchyshows\; that\; the\; true\; theory\; of\; second-order\; arithmetic\; is\; not\; definable\; by\; any\; single\; formula\; in\; second-order\; arithmetic.$$

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