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  "body": "\n## Definition\n\nLet $\\mathsf{RCA}_0$ be the subsystem of [[second-order arithmetic|DC60FYCSPM]] with the basic axioms, induction for $\\Sigma^0_1$ formulas, and comprehension for $\\Delta^0_1$ formulas.\nA sentence $\\tau$ of second-order arithmetic is *equivalent over* $\\mathsf{RCA}_0$ to a subsystem $S$ if:\n\n1. $S$ proves $\\tau$;\n2. $\\mathsf{RCA}_0 + \\tau$ proves every axiom of $S$.\n\n*Reverse mathematics* is the study of which theorems of ordinary mathematics are provable in which subsystems of second-order arithmetic.\nIn particular, it studies equivalences over $\\mathsf{RCA}_0$.\n\n## Remarks\n\nThe big five subsystems are $\\mathsf{RCA}_0$, $\\mathsf{WKL}_0$ (adds weak König's lemma), $\\mathsf{ACA}_0$ (arithmetical comprehension), $\\mathsf{ATR}_0$ (arithmetical transfinite recursion) and $\\Pi^1_1\\text{-}\\mathsf{CA}_0$ ($\\Pi^1_1$ comprehension), each stronger than the one before.\n\nExample: the Bolzano–Weierstrass theorem for sequences in $[0,1]$ is equivalent to $\\mathsf{ACA}_0$ over $\\mathsf{RCA}_0$.\nThe Heine–Borel theorem for countable covers of $[0,1]$ by open intervals is equivalent to $\\mathsf{WKL}_0$ over $\\mathsf{RCA}_0$.\n\nNon-example: weak König's lemma is provable in $\\mathsf{ACA}_0$ but not equivalent to $\\mathsf{ACA}_0$ over $\\mathsf{RCA}_0$.\nClause 2 fails, because $\\mathsf{WKL}_0$ does not prove arithmetical comprehension.\n\nThe models of $\\mathsf{RCA}_0$ with the standard natural numbers include the one whose sets are the computable sets, so $\\mathsf{RCA}_0$ corresponds to computable mathematics.\n",
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