Editing Reverse mathematics (section)

===Stronger systems===
Over RCA<sub>0</sub>, '''Π{{su|p=1|b=1}}''' transfinite recursion, '''∆{{su|p=0|b=2}}''' determinacy, and the '''∆{{su|p=1|b=1}}''' Ramsey theorem are all equivalent to each other.

Over RCA<sub>0</sub>, '''Σ{{su|p=1|b=1}}''' monotonic induction, '''Σ{{su|p=0|b=2}}''' determinacy, and the '''Σ{{su|p=1|b=1}}''' Ramsey theorem are all equivalent to each other.

The following are equivalent:<ref>{{cite conference |conference=LICS '16: 31st Annual ACM/IEEE Symposium on Logic in Computer Science |date=2016  |title=How unprovable is Rabin's decidability theorem? |first1=Leszek |last1=Kołodziejczyk |first2=Henryk |last2=Michalewski |arxiv=1508.06780 }}</ref><ref>{{cite web |last=Kołodziejczyk |first=Leszek |url = https://cs.nyu.edu/pipermail/fom/2015-October/019257.html |title=Question on Decidability of S2S |publisher=FOM |date=October 19, 2015}}</ref>
* (schema) Π{{su|p=1|b=3}} consequences of Π{{su|p=1|b=2}}-CA<sub>0</sub>
* RCA<sub>0</sub> + (schema over finite ''n'')  determinacy in the ''n''th level of the difference hierarchy of '''Σ{{su|p=0|b=2}}''' sets
* RCA<sub>0</sub> + {τ: τ is a true [[S2S (mathematics)|S2S]] sentence}

The set of Π{{su|p=1|b=3}} consequences of second-order arithmetic Z<sub>2</sub> has the same theory as RCA<sub>0</sub> + (schema over finite ''n'')  determinacy in the ''n''th level of the difference hierarchy of '''Σ{{su|p=0|b=3}}''' sets.<ref>{{cite journal |first1=Antonio |last1=Montalban |first2=Richard |last2=Shore |title=The limits of determinacy in second order arithmetic: consistency and complexity strength | journal=[[Israel Journal of Mathematics]] |volume=204 |year=2014 | pages=477–508 |doi=10.1007/s11856-014-1117-9 | doi-access=|s2cid=287519 }}</ref>

For a [[poset]] <math>P</math>, let <math>\textrm{MF}(P)</math> denote the topological space consisting of the filters on <math>P</math> whose open sets are the sets of the form <math>\{F\in\textrm{MF}(P)\mid p\in F\}</math> for some <math>p\in P</math>. The following statement is equivalent to <math>\Pi^1_2\mathsf{-CA}_0</math> over <math>\Pi^1_1\mathsf{-CA}_0</math>: for any countable poset <math>P</math>, the topological space <math>\textrm{MF}(P)</math> is [[Completely metrizable space|completely metrizable]] iff it is [[Regular topological space|regular]].<ref>C. Mummert, S. G. Simpson. "Reverse mathematics and <math>\Pi^1_2</math> comprehension". In ''Bulletin of Symbolic Logic'' vol. 11 (2005), pp.526–533.</ref>