Reverse Mathematics and Π¹₂ Comprehension

Abstract

Abstract. We initiate the reverse m athem atics of generaltopology. We show thata certain m etrization theorem is equivalent to Π 1 2 com prehension. An MF space is defi ned to be a topologicalspace of the form MF(P) with the topology generated by {Np | p ∈ P}. Here P is a poset, MF(P) is the setof m axim alfilters on P, and Np = {F ∈ MF(P) | p ∈ F }.If the poset P is countable, the space MF(P) is said to be countably based. The cl ass of countably based MF spaces can be defi ned and discussed within the subsystem ACA 0 of second order arithm etic. One can prove within ACA 0 that every com plete separable m etric space is hom eom orphic to a countably based MF space which is regular. We show that the converse statem ent, "every countably based MF space which is regularis hom eom orphicto a com plete separable m etric space," is equivalent to Π 1 2 -CA 0 . The equivalence is proved in the weaker system Π 1 1 -CA 0 . This is the fi rstexam ple of a theorem of core m athem atics which is provable in second orderarithm etic and im plies Π 1 2 com prehension. In the foundations of m athem atics, there is an ongoing research program

Reverse Mathematics of Mf Spaces

“…In this section, we show in Π 1 1 -CA 0 that Π 1 2 comprehension is required to prove this result. A sketch of the proof may be found in [14]. A complete proof appeared in the author's PhD thesis [13].…”

The limits of determinacy in second order arithmetic: consistency and complexity strength

“…In fact, for those theorems that are not true effectively (essentially the same as provable in RCA 0 ), the vast majority turn out to be equivalent in the sense of reverse mathematics to one of the two weakest of these systems (WKL 0 or ACA 0 ). Only a handful are equivalent to one of the two stronger systems (ATR 0 or Π (Mummert and Simpson [2005]) to be at the next level (Π 1 2 -CA 0 ) and (as far as we know) none that are not known to be provable a bit beyond this level but still at one less than Π 1 3 -CA 0 . In this paper we supply a natural hierarchy of theorems that require, respectively, each of the natural levels of set existence assumptions going from the previous known systems all the way up to full second order arithmetic.…”

Set Existence Principles and Closure Conditions: Unravelling the Standard View of Reverse Mathematics†

“…In particular, Π 1 1 -CA 0 is not equivalent over ATR 0 to any Π 1 2 statement, although it is straightforwardly expressed as a Π 1 3 sentence. There are even theorems that exceed the strength of Π 1 1 -comprehension, such as "Every countably based MF space which is regular is homeomorphic to a complete separable metric space", which is equivalent to Π 1 2 -comprehension [Mummert and Simpson, 2005]. Such theorems are not expressible as Π 1 3 statements, so we must consider yet more complex sentences as also expressing closure conditions if we are to bring them into the account.…”