Titles and Abstracts

More titles/abstracts will be added as we receive them.

Andrew Arana: Varieties of reversals

Peter Cholak: On Ramsey-like theorems on the rationals and the Rado graph

Ramsey theorem implies for every 2-coloring of pairs of naturals there
is an infinite set H of naturals where all the pairs formed from H
have the same color. We will explore how to extend this to the
rationals, the Rado graph and some other relational homogenous
countable structures. One of the main tools used in these extensions
is Milliken's tree theorem and it’s recent modifications. Our goal is
to try to understand the arithmetic complexity of the resulting
“homogenous” set or structure. I.e. if the colorings are computable is
there a “homogenous” structure within the arithmetical hierarchy and
if so where.


Benedict Eastaugh: On the indispensability of mathematics to philosophy

Many arguments in contemporary philosophy rely on substantial
mathematical results. For example, Dutch book theorems are used to
argue that our degrees of belief ought to obey the probability axioms,
while Arrow's theorem has been used to argue that true democracy is
impossible. But despite the centrality of mathematics to their work,
philosophers working in core areas such as metaphysics, epistemology,
and ethics often see their subjects as disconnected from foundational
concerns about mathematics itself. In fact, it is not uncommon for
people working in these areas to simultaneously employ mathematical
techniques while also expressing sympathy for nominalism (according to
which mathematical objects don't exist) or logicism (according to
which mathematical axioms need to be justified on a purely logical
basis). In this talk, we will use the tools of reverse mathematics to
argue that this dissonance between the philosophical applications of
mathematical results and the rejection of mathematics as a substantive
subject matter is unsustainable. To support this claim, we will
examine some classical arguments from the philosophy of logic and
political philosophy which we will argue rely on the truth of
mathematical principles like weak König's lemma and arithmetical
comprehension. We will accordingly suggest that reverse mathematics
provides a powerful (but as yet largely untapped) means of
understanding the ways in which mathematics is indispensable to
philosophy.


David Fernandez-Duque: Noetherian Gödel logics

Joint work with Juan P. Aguilera and Jan Bydzovsky

Noetherian Gödel logics are many-valued logics where the set of truth
values is a closed subset of [0,1] inversely isomorphic to a countable
ordinal. In this talk we discuss the complexity of satisfiability and
validity for each Noetherian Gödel logic, strengthening and generalizing
results of Baaz-Leitsch-Zach and Hájek. Specifically, we characterize
each decision problem in terms of truth complexity over suitable stages
of the constructible hierarchy.


Harvey Friedman: Strict reverse mathematics

Reverse Mathematics is almost entirely based on the base theory RCA0 I
introduced at its founding. Reverse Mathematics was preceded by an
earlier initiative that I call Strict Reverse Mathematics, with the
idea that all reversals are of small collections of purely
mathematical statements, ideally pulled from the mathematical
literature. We have established equivalences of the main systems of RM
with such collections, and propose associated research programs that
go well beyond the current RM in breadth and scope. Considerable
discussion can be found at
https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2021/12/RMfoundingETF122921a.pdf


Hugo Herbelin: Constructive reverse mathematics of Gödel's completeness theorem

Gödel's completeness theorem has been studied in different forms in
constructive and classical reverse mathematics in either set theory or
second order arithmetic, where it is connected alternatively to the
ultrafilter lemma, the weak Fan theorem or weak König's lemma,
themselves actually occurring in different forms of different
intuitionistic strength. We will give a unified overview of these
results, highlighing the impact of the presence of specific
connectives on the intuitionistic strength of the completeness
statement: Markov's principle is required when the false connective is
present while disjunction relates to Double Negation Shift. We will
also sketch how reverse mathematics in linear logic might help to
factorize constructive and classical reverse mathematics.

Antonio Montalban: The limit of detereminacy in second order logic

Carl Mummert:: Coding in reverse mathematics 

Leonardo Pacheco / Keita Yokoyama: Determinacy and reflection principles in second-order arithmetic

It is well-known that several variations of the axiom of determinacy
play important roles in the study of reverse mathematics, and the
relation between the hierarchy of determinacy (especially the level of
Sigma^0_2 and Sigma^0_3) and comprehension axioms are revealed by
Tanaka, Nemoto, Montalbán, Shore, and others. In this talk, we show
variations of a result by Kołodziejczyk and Michalewski relating
determinacy and reflection in second-order arithmetic based on a
model-theoretic characterization of the reflection principles.


Fedor Pakhomov / Andreas Weiermann: Functorial Fast-Growing Hierarchies

Fast-growing hierarchies are sequences of functions obtained through
various processes similar to the ones that yield multiplication from
addition, exponentiation from multiplication, etc. We observe that
fast-growing hierarchies can be naturally extended to functors on the
categories of natural numbers and of linear orders. We show that the
categorical extensions of binary fast-growing hierarchies to ordinals
are isomorphic to denotation systems given by ordinal collapsing
functions, thus establishing a connection between two fundamental
concepts in Proof Theory. Using this fact, we obtain a restatement of
the subsystem Π11-CA0 of analysis as a higher-type wellordering
principle.

Victor Pambuccian: Reverse geometry

Sam Sanders: Big in RM: the uncountability of R

That the infinite comes in different ’sizes’ is a relatively new
insight, established by Cantor in the first set theory paper (1874) in
the guise of the uncountability of R.  We develop the higher-order RM
of the latter, focussing on the following points:

a) the usual definition of 'countable set' (based on
injections/bijections to N) is unsuitable for our RM-study: countable
sets that occur ‘in the wild’ require a different (equivalent over
strong systems) definition, also stemming from the literature.

b) with the new definition of `countable set’ from a), we observe that
the uncountability of R is another ‘Big’ system of RM, with
equivalences involving the Riemann integral, Volterra’s early work,
and related topics.

c) The RM of countable sets in general yields more ‘Big’ systems,
including the study of regulated and bounded variation functions and
their well-known properties.

We discuss the myriad implications for the philosophy of mathematics
of our results, esp.  the claimed linearity of Simpson’s Goedel
hierarchy.

Paul Shafer: The logical and computational strength of inside/outside Ramsey theorems

Rival and Sands proved that every infinite graph G contains an
infinite subset H such that every vertex of G is adjacent to precisely
none, one, or infinitely many vertices of H.  We call this result an
inside/outside Ramsey theorem because the conclusion provides
information about vertices that are inside of H and about vertices
that are outside of H.  Rival and Sands also proved a similar
statement for infinite partial orders of finite width.  We analyze the
strength of these theorems from the perspective of reverse mathematics
and the Weihrauch degrees.  We find that they give the first examples
from the modern general mathematics literature of theorems that are
equivalent to the double jump of weak König's lemma in the Weihrauch
degrees and of theorems that are equivalent to the
ascending/descending sequence principle (plus Sigma_2 induction in
some cases) in reverse mathematics.  This work is joint with Marta
Fiori Carones, Alberto Marcone, and Giovanni Soldà.

Theodore Slaman: On Cauchy Sequences