Carl Mummert

Reverse mathematics of MF spaces

This paper gives a formalization of general topology in second-order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology. For each poset P we let MF(P) denote the set of maximal filters on P endowed with the topology generated by {Np | p ∈ P}, where Np = {F ∈ MF(P) | p ∈ F}. We define a countably based MF space to be a space of the form MF(P) for some countable poset P. The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces. The following reverse mathematics results are obtained. The proposition that every nonempty Gδ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to over ACA0. The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over to the proposition that is countable for all A ⊆ ℕ. The proposition that every regular countably based MF space is homeomorphic to a complete separable metric space is equivalent to over .

On the strength of the finite intersection principle

We study the logical content of several maximality principles related to the finite intersection principle (FIP) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to ACA0 over RCA0, while others are strictly weaker and incomparable with WKL0. We show that there is a computable instance of FIP every solution of which has hyperimmune degree, and that every computable instance has a solution in every nonzero c.e. degree. In particular, FIP implies the omitting partial types principle (OPT) over RCA0. We also show that, modulo Σ20 induction, FIP lies strictly below the atomic model theorem (AMT).

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a

Stationary and convergent strategies in Choquet games

\Pi^1_2

Topological aspects of poset spaces

sentence of a certain form is provable using E-HA

Reverse Mathematics of Matroids

{}^\omega

On the existence of a connected component of a graph

along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the \textit{Dialectica} interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.

Reverse mathematics and properties of finite character

If Nonempty has a winning strategy against Empty in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows Nonempty to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits Nonempty to consider the previous move by Empty. We show that Nonempty has a stationary winning strategy for every second countable T1 Choquet space. More generally, Nonempty has a stationary winning strategy for any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. A T1 space X is the open continuous image of a complete metric space if and only if Nonempty has a convergent winning strategy in the Choquet game on X. A T1 space X is the open continuous compact image of a metric space if and only if X is metacompact and Nonempty has a stationary convergent strategy in the Choquet game on X. A T1 space X is the open continuous compact image of a complete metric space if and only if X is metacompact and Nonempty has a stationary convergent winning strategy in the Choquet game on X.

The modal logic of Reverse Mathematics

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.

Filters on Computable Posets

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for