23 research outputs found
Topological aspects of poset spaces
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.Comment: 29 pages. To be published in the Michigan Mathematical Journa
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
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 sentence of a certain form is provable using E-HA
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.Comment: Accepted, Notre Dame Journal of Formal Logi
Stationary and convergent strategies in Choquet games
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. (1) A T1 space X is the open image of a complete metric
space if and only if NONEMPTY has a convergent winning strategy in the Choquet
game on X. (2) A T1 space X is the compact open 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. (3) A T1 space X is the compact open 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.Comment: 24 page
Reverse mathematics and properties of finite character
We study the reverse mathematics of the principle stating that, for every
property of finite character, every set has a maximal subset satisfying the
property. In the context of set theory, this variant of Tukey's lemma is
equivalent to the axiom of choice. We study its behavior in the context of
second-order arithmetic, where it applies to sets of natural numbers only, and
give a full characterization of its strength in terms of the quantifier
structure of the formula defining the property. We then study the interaction
between properties of finite character and finitary closure operators, and the
interaction between these properties and a class of nondeterministic closure
operators.Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse
mathematics and equivalents of the axiom of choice", which has been
abbreviated and divided into two pieces for publicatio
The modal logic of Reverse Mathematics
The implication relationship between subsystems in Reverse Mathematics has an
underlying logic, which can be used to deduce certain new Reverse Mathematics
results from existing ones in a routine way. We use techniques of modal logic
to formalize the logic of Reverse Mathematics into a system that we name
s-logic. We argue that s-logic captures precisely the "logical" content of the
implication and nonimplication relations between subsystems in Reverse
Mathematics. We present a sound, complete, decidable, and compact tableau-style
deductive system for s-logic, and explore in detail two fragments that are
particularly relevant to Reverse Mathematics practice and automated theorem
proving of Reverse Mathematics results
Banach's theorem in higher order reverse mathematics
In this paper, methods of second order and higher order reverse mathematics
are applied to versions of a theorem of Banach that extends the
Schroeder-Bernstein theorem. Some additional results address statements in
higher order arithmetic formalizing the uncountability of the power set of the
natural numbers. In general, the formalizations of higher order principles here
have a Skolemized form asserting the existence of functionals that solve
problems uniformly. This facilitates proofs of reversals in axiom systems with
restricted choice.Comment: Version: count-20230393arxiv.tex adding publication info in header
and listing modification