The system ACA 0 is a conservative extension of "'first-order arithmetic "'( or first-order Peano axioms ), defined as the basic axioms, plus the first order induction axiom scheme ( for all formulas ? involving no class variables at all, bound or otherwise ), in the language of first order arithmetic ( which does not permit class variables at all ).
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The system ATR 0 adds to ACA 0 an axiom which states, informally, that any arithmetical functional ( meaning any arithmetical formula with a free number variable " n " and a free class variable " X ", seen as the operator taking " X " to the set of " n " satisfying the formula ) can be iterated transfinitely along any countable well ordering starting with any set.
