Editing Reverse mathematics (section)

=== Arithmetical transfinite recursion ATR<sub>0</sub>===

The system ATR<sub>0</sub> adds to ACA<sub>0</sub> an axiom that states, informally, that any arithmetical functional (meaning any arithmetical formula with a free number variable ''n'' and a free set 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.  ATR<sub>0</sub> is equivalent over ACA<sub>0</sub> to the principle of &Sigma;{{su|p=1|b=1}} separation. ATR<sub>0</sub> is impredicative, and has the [[Ordinal analysis|proof-theoretic ordinal]] <math>\Gamma_0</math>, the supremum of that of predicative systems.

ATR<sub>0</sub> proves the consistency of ACA<sub>0</sub>, and thus by [[Gödel's incompleteness theorems|Gödel's theorem]] it is strictly stronger.

The following assertions are equivalent to
ATR<sub>0</sub> over RCA<sub>0</sub>:
* Any two countable well orderings are comparable. That is, they are isomorphic or one is isomorphic to a proper initial segment of the other.<ref name="Simpson2009" /><sup>theorem V.6.8</sup>
* [[Ulm's theorem]] for countable reduced Abelian groups.
* The [[perfect set property|perfect set theorem]], which states that every uncountable closed subset of a complete separable metric space contains a perfect closed set.
* [[Lusin's separation theorem]] (essentially &Sigma;{{su|p=1|b=1}} separation).<ref name="Simpson2009" /><sup>Theorem V.5.1</sup>
* [[Determinacy]] for [[open set]]s in the [[Baire space]].