Abstracts of Keynote Lectures
 Matthias Aschenbrenner.
tba
 Manuel Bodirsky.
tba
 Stefania Centrone.
tba
 Victoria Gitman.
Choice schemes for KelleyMorse set theory
Abstract:
The defining feature of the secondorder KelleyMorse (KM) set theory is that its class comprehension scheme includes
all secondorder assertions. The full choice scheme includes, for every secondorder formula $\phi$, the assertion that
if for every set $x$, there is a class $Y$ such that $\phi(x,Y)$ holds, then there is a single class $Z$ that has among
its slices a witnessing class for every $x$. The choice scheme can be stratified by formula complexity and has obvious
setsized variants. Choice schemes for KM were first studied by Mostowski and Marek. They have been considered in the
context of nonstandard set theory and more recently in analyzing class forcing extensions. I will show that even the
weakest choice scheme, namely choice for omega and formulas with no secondorder quantifiers, is already independent of KM.
This provides an interesting contrast with the situation in secondorder arithmetic, where $Z_2$, the analogue of KM,
proves all instances of $\Sigma^1_3$ choice, but the choice scheme may fail at the $\Sigma^1_3$ level. I will also show
that the full choice scheme does not follow from the setsized version. Finally, I will argue that KM+, which is KM
together with the choice scheme, is a more robust theory than KM because it is preserved by ultrapowers of the universe
and proves that secondorder complexity classes are closed under firstorder quantification, while it is consistent that
ultrapowers of KM models fail to even satisfy KM and that in a model of KM, secondorder complexity classes can fail to
be closed under even bounded firstorder quantification. This is joint work with Joel David Hamkins and Thomas Johnstone.
 Fairouz Kamareddine.
tba
 Anca Muscholl.
Distributed synthesis
Abstract:
Synthesis is about constructing systems that behave correctly w.r.t. to a given specification.
Synthesizing distributed systems is an attractive goal, since such systems are notoriously difficult to get right.
Unfortunately, there are very few known decidable frameworks for distributed synthesis.
We will discuss one such framework, stated in the rendezvous model of Zielonka automata. In this setting,
synthesis is decidable for all omegaregular local specifications, under the restriction that the communication
graph of the system is acyclic.
 Olivier Roy.
tba
 Daniel Turetsky.
tba
Abstracts of PhD Colloquium Lectures
 Dominik Adolf.
tba
 Marcos Kramer.
The Naproche system: Proofchecking mathematical texts in controlled natural language
Abstract:
We have developed a controlled natural language (CNL) – i.e. a subset of a natural language defined through a formal grammar –
for writing mathematical texts. The Naproche system can check the correctness of mathematical proofs written in this CNL.
In this talk we will highlight some interesting logical and linguistic problems that had to be solved in the development of
the Naproche CNL and the Naproche system: The system uses a formalism from the linguistic theory of presuppositions in order
to work with potentially undefined terms, and can handle dynamic quantification, including the phenomenon of implicit dynamic
function introduction, which had previously not been discussed in the literature.
 Tomer Kotek.
tba
 Davide Rinaldi.
tba
Abstracts of Contributed Talks
 Geghard Bedrosian and Frederik Herzberg.
Ultraproduct construction of representative utility functions with infinitedimensional domain
Abstract: This
paper applies model theory to macroeconomic theory. In mathematical
models of macroeconomic theory, the hypothesis of a "representative
agent'' is ubiquitous, but the search for a rigorous justification has
so far been unsuccessful and was ultimately abandoned until very
recently. Herzberg (2010) constructed a representative utility function
for finitedimensional social decision problems, based on an bounded
ultrapower construction over the reals, with respect to the ultrafilter
induced by the underlying social choice function (via the
KirmanSondermann (1972) correspondence). However, since the decision
problems of macroeconomic theory are typically infinitedimensional,
Herzberg's original result is insufficient for many applications. We
therefore generalise his result by allowing the social alternatives to
belong to a general reflexive Banach space $W$; in addition to known
results from convex analysis, our proof uses a nonstandard enlargement
of the superstructur eover $W\cup\mathbb{R}$, obtained by a bounded
ultrapower construction with respect to the KirmanSondermann
ultrafilter.
 Alexander Carstensen Block.
The Modal Logic of Symmetric Extensions
Abstract: Symmetric
forcing was first used by Paul Cohen to show the independence of the
Axiom of Choice from $\mathsf{ZF}$. It is a modification of the method of forcing, a
flexible set theoretic model construction assigning to a given
(countable transitive) model of $\mathsf{ZF}$ outer models, called generic
extensions, each of which is "generated" from a single new object. The
forcing method itself is unsuitable for establishing the independence of
the Axiom of Choice from $\mathsf{ZF}$, since every generic extension of a
$\mathsf{ZFC}$model also satisfies the Axiom of Choice.
So instead Cohen used symmetric forcing, considering symmetric
submodels of generic extensions instead of actual generic extensions.
Symmetric submodels are obtained by deleting objects whose descriptions
(in terms of the socalled forcing language) are not sufficiently stable
under a certain group action. This process may remove wellorderings
of sets still present, thereby breaking the axiom of choice.
We refer to symmetric submodels of generic extensions in short as
symmetric extensions. A central property of symmetric extensions (and
generic extensions as well) is that truth in them can be expressed in
the ground model one starts with. More precisely for every firstorder
sentence $\varphi$ in the language of set theory there is a firstorder sentence
$\varphi^*$ in the language of set theory such that for any (countable
transitive) $\mathsf{ZF}$model $M$ we have that $M \models \varphi^*$ iff for any symmetric
extension $N$ of $M$: $N$ satisfies $\varphi$.
Using this fact we can give a correspondence between formulas in the
basic modal language and firstorder sentences in the language of set
theory as follows. A translation is a map assigning proposition letters
to sentences in the language of set theory. We can extend any
translation $T$ to an interpretation $\underline{T}$ assigning formulas in the basic
modal language to firstorder sentences in the language of set theory by
recursively setting:
(1) $\underline{T}(p) := T(p)$ for any proposition letter $p$;
(2) $\underline{T}(\bot) := (0=1)$;
(3) $\underline{T}(A+B) := \underline{T}(A) + \underline{T}(B)$ for any logical connective $+$;
(4) $\underline{T}(\Box A) := [\underline{T}(A)]^*$
With this we can define the ZFprovable modal logic MLS of symmetric
forcing by setting $MLS := \{A \mid ZF$ proves $\underline{T}(A)$ for any translation $T\}$
This definition is analogous to the definitions of the corresponding
modal logic of forcing as introduced and investigated by Joel Hamkins
and Benedikt Löwe in [1]. Using an adaptation of their methods we show
that MLS = S4.2 and therefore that MLS coincides with the modal logic of
forcing.
References:
1. JOEL D. HAMKINS, BENEDIKT LÖWE, The modal logic of forcing.
Transactions of the American Mathematical Society, vol. 360 (2008), no.
4, pp. 17931817
 Vasco Brattka, Guido Gherardi and Rupert Hölzl.
Probabilistic computability and choice
Abstract: The
aim of this talk is to give an overview of the systematic investigation
we have carried out onto the field of randomized algorithms for real
numbers.
In particular, the notions of Las Vegas and probabilistic computability for infinite sequences are presented.
As case studies we analyze the problems of computing Nash equilibria and solutions to the intermediate value theorem.
See the arXiv version for more details.
 Yong Cheng, SyDavid Friedman and Joel David Hamkins.
Large cardinals need not be large in HOD
Abstract:
We
prove that large cardinals need not generally exhibit their large
cardinal nature in HOD. For example, a supercompact cardinal \kappa need
not be weakly compact in HOD, and there can be a proper class of
supercompact cardinals in V, none of them weakly compact in HOD, with no
supercompact cardinals in HOD. Similar results hold for many other
types of large cardinals, such as measurable and strong cardinals.
 Merlin Carl.
Algorithmic Randomness for Infinitary Machines
Abstract: During
the last decade, a number of machine models of transfinite
computability have been proposed and studied, such as Infinite Time
Turing Machines, Infinite Time Register Machines and Ordinal Turing
Machines.
We consider algorithmic randomness for machine models of infinitary
computations. We show that a theorem of Sacks, according to which a real
$x$ is computable from a randomly chosen real with positive probability
iff it is recursive holds for many of these models, but is independent
from $ZFC$ for ordinal Turing machines. Furthermore, we define an
analogue of MartinLöfrandomness for Infinite Time Register Machines
and show that some classical results like van Lambalgen's theorem
continue to hold.
 Bernhard Fisseni. What makes stories similar? – Some Empirical and Formal Results
Abstract: We
present a survey of the results and findings of the research project
*What makes stories similar?* funded by the John Templeton Foundation
from October 2011 to May 2015.
The research project *What makes stories similar?* funded by the
John Templeton Foundation from October 2011 to May 2015 and coordinated
by Benedikt Löwe as principal investigator aimed at providing a
discussion of similarity for narratives. Guided by its eponymous
question and based on Löwe's previous research on comparability and
(in)compatibility of formal frameworks, the project aimed to find out
whether there are structural (rather than presentational) properties
that contribute or even define story similarity, whether they can be
expressed in formal representation systems, and how such representations
can be empirically tested.
 Walter Hoering. A Formalism allowing Nondeterministic Strategies
Abstract:
Turing and Feferman have shown, how to interpret and use Gödel's incompleteness
theorem in a positive way  i.e. as a tool to construct progressions of recursively
enumerable theories. These theories work with constructive ordinals
and are either effectively realizable and recursively enumerable,
therefore Gödelextendible, using as they do, only effective rules and
only a recursively enumerable part of the constructive ordinals, or they
reach beyond he effective, using some version of the omegarule or a
nontrivial section of the constructive ordinals.
If one deals with these progressions, one can work within a conventional system or
abandon it fore another one, which may be stronger. This
unconditioned and permanent capacity for choice is beyond the capacities
of a conventional formal system. At this point we cannot but ask for
the difference between the capacities of a human being and the
possibilities of a conventional formal system. The answer is quite
simple: the difference lies in the freedom of man, in his capacity for
random choice. So we come to look for a new kind of formal system
featuring this capacity. We construct a new kind of formal system, the
nondeterministic systems which have nondeterminism built in in a
presumably novel way. It seems to go beyond the recursively enumerable.
We call a system nondeterministic, if it can take a random input or
alternately also a deterministic input at the same place. Some of these
systems are capable of strengthening themselves indefinitely by dint of
their own strength, just as Baron Münchhausen pulled himself and the
horse upon which he was riding out of a pernicious swamp just by pulling
up his own tuft of hair. Therefore we propose the name of
MünchhausenSystems for this kind of system. Another adventure of the
Baron was that, riding on a canon ball, he suddenly decided to change
direction by jumping onto another canon ball approaching from the
opposite direction. Also this story will have a counterpart here. Only
that the rather pedestrian analogue of the Baron's instantaneous
resolution, a random generator, will be used to create an infinite
sequence of natural numbers and thereby of systems.
Turing in his conception of a device, later to be called Turing
machine, dealt with linear computation, proceeding without branching:.
"There is only one way to go on, straight ahead".
Housewives and artisans know: If I do one thing, I will have to skip something
other. It is this kind of inescapable choice between many alternatives, which we model in our systems.
Our subject is computability, deducibility if free, nondetermined
decisions are allowed to intervene an infinite number of times: "There
are always many directions to go on" In our Münchhausen Systems
(Msystems for short) we always face the alternative: to stay where we
are and to continue working in the present surrounding  or  to change
into a new environment.
If one is interested in evaluating sets like O, Kleene's
constructive ordinals, or B the set of recursive wellfounded trees one
deals with positive inductive definitions I.e. with productive sets  to
which MSystems can be applied.
 Mirjana Ilic and Branislav Boricic.
Is t really needed in formulating sequent calculi for relevant logics?
Abstract: The
first Gentzen calculi for distributive relevant logics were presented
in the late 1960's and early 1970's, when Dunn published Gentzen system
for positive relevant logic, and Minc formulated the Gentzen system
for positive relevant logic with necessity. In those calculi, an empty
lefthand side, in the left premise of the cut rule can lead to
irrelevance (the modal fallacy would be derivable, then).
There are two ways to prevent the inference of the modal fallacy in
those calculi. However in both of them, the constant t, which is not
the part of the standard language for relevant logics, is needed.
Giambrone and Brady also followed those ideas in formulating the
sequent calculi for RW+ and RW, confirming the need for t in those
logics, also. However, it has been shown recently, that it is possible
to obtain the cutfree sequent calculus for RW without the help of t.
We present the sequent calculus GRW of RW. The calculus GRW is with
righthanded sequents, based on unsigned formulae, unlike Brady's
calculus L'RW of RW, which is based on singleconclusion sequents with
signed formulae. The calculus GRW has a much neater presentation,
although the proof that it is cutfree is somewhat more complex.
 Franziska Jahnke.
Definable Henselian Valuations
Abstract: In
a henselian valued field, many arithmetic or algebraic questions can be
reduced, via the henselian valuation, to simpler questions about the
value group and the residue field. By the celebrated AxKochen/Ershov
Principle, in fact, if the residue characteristic is 0, `everything' can
be so reduced:
the 1storder theory of a henselian valued field (as a valued field)
is fully determined by the 1storder theory of its value group (as an
ordered abelian group) and of its residue field (as a pure field).
Conversely, in many cases, a henselian valuation is so intrinsic to a
field that it is itself encoded in the arithmetic of the field, i.e.
its valuation ring is a 1storder parameterfree definable subset of the
field (in the language of rings). In this talk, we investigate the
question when a henselian valued field admits such a nontrivial
0definable henselian valuation.
Clearly, separably or real closed fields admit no nontrivial
definable valuations, and, by the work of Prestel and Ziegler, there are
further examples of henselian valued fields which do not admit a
0definable nontrivial henselian valuation. We give conditions on the
residue field which ensure the existence of a parameterfree definiton.
Using recent work of Prestel, we deduce that in every case discussed in
the talk the formula can be found to be either existentialuniversal or
universalexistential.
The material presented includes joint work with both Jochen Koenigsmann and Arno Fehm.
 Andreas Kapsner.
An Expressibility Problem for Constructive Logics
Abstract: This paper discusses a problem in M. Dummett’s argument for intuitionistic
logic raised by T. Williamson, who subsequently to raising it advocated a change
of the account of negation in intuitionistic logic to solve it. I will show that it
is actually questionable how well this change deals with the problem.
To sketch the problem, take the example Williamson discussed in [Williamson(1994)],
Goldbach’s conjecture. Dummett wants to say of Goldbach’s conjecture that
it may never be decided, else his argument for intuitionistic logic will not get
off the ground. Williamson argues that this claim is actually intuitionistically
inconsistent.
It is important that the problem Williamson points out is not with the claim
that it might be proved that neither Goldbach’s conjecture nor its negation can
be proved. He has a claim in his sights that would seem to be much
less controversially acceptable to all parties involved: That it might
happen that when
the last mathematicians have breathed their last breaths Goldbach’s conjecture
stands unresolved. In this scenario, no one will have managed to prove either
Goldbach’s conjecture or its negation. However, also no proof of its undecidability
in principle will have been discovered, either because such a proof is
impossible or because, again, simply no one managed to find the proof.
In addition to giving a detailed analysis of the problem he raises, Williamson
suggest a solution to the problem. This solution involves changing
the intuitionistic account of negation. Instead of saying that notA is
verified by showing that a verification of A is impossible, we should
ask for something stronger.
To verify notA, we need to be able to falsify A. This idea can be funneled into
a logical system quite nicely, and one ends up with what is known as Nelson
logic (cf. [Nelson(1949)]). Williamson claims that this move will
solve the problem he raised, that is, assuming Nelson logic, we can say
that there are undecidable statements.
However, I will show that this is not true in the sense he means it. Employing
the formal analysis of the problem he chose and the usual Kripke
semantics for Nelson logic, one can come to see that it is still
impossible to assert the
following: We might, just due to bad luck and a lack of inspired ideas, never
decide Goldbach’s conjecture. Intriguingly, what does become possible is to
assert the seemingly stronger claim that we might find out the impossibility
of deciding Goldbach’s conjecture. That is, we can say that there might be
statements for which we’ll learn that we will never verify or falsify them. But
the case Williamson had in mind, where we would like to say that it is possible
that we may simply never find a proof without knowing that we won’t, can not
be accounted for. The paper ends by outlining some morals we might draw from
this problematic situation.
References
[Nelson(1949)] D. Nelson. Constructible falsity. Journal of Symbolic logic, 14/1:16–26, 1949.
[Williamson(1994)] T. Williamson. Never say never. Topoi, 13(2):135–145, 1994.
 Yurii Khomskii.
Fullsplitting Miller trees and infinitely often equal reals
Abstract: We
investigate two closely related partial orders of trees on the Baire
space: the "fullsplitting MIller trees" and the "infinitely often equal
trees" as well as their corresponding sigmaideals. The former notion
was considered by Newelski and Roslanowski while the latter involves a
correction of a result of Spinas. We consider some Marczewskistyle
regularity properties based on these trees, which turn out to be closely
related to the property of Baire, and look at the dichotomies of
NewelskiRoslanowski and Spinas for higher projective pointclasses. We
also provide some insight concerning a question of Fremlin whether one
can add an infinitely often equal real without adding a Cohen real.
This is joint work with Giorgio Laguzzi (Hamburg).
 Takayuki Kihara and Arno Pauly.
Point degree spectra of represented spaces
Abstract: We
introduce the point spectrum of a represented spaces as a substructure
of the Medvedev degrees. The point spectrum is closely linked to
isomorphism type of a space w.r.t. countably continuous maps, and via
this, also with the dimension. Through this new connection between
descriptive set theory and degree theory (as part of computability
theory) we can answer several open questions.
As a result on the way, we prove that any admissible represented
space with an effectively fibercompact representation is already
computably metrizable.
 Giorgio Laguzzi.
Roslanowski and Spinas dichotomies
Abstract: (the talk is bases on joint works with Yurii Khomskii and Wolfgang Wohofsky)
We investigate two tree forcings for adding infinitely often equal
reals: the full splitting Miller forcing $\mathbb{FM}$, introduced by
Roslanowski in [1], and the infinitely often equal trees
forcing $\mathbb{IE}$, implicitly introduced by Spinas in [2].
We prove results about Marczewskitype regularity properties associated
with these forcings as well as dichotomy properties on
$\mathbf{\Delta}^1_2$ and $\mathbf{\Sigma}^1_2$ levels, with a
particular emphasis on a parallel with the Baire property. Furthermore,
we prove that our dichotomies hold for all projective sets in Solovay's
model, and that the use of an inaccessible is necessary for both.
Finally we also investigate the Marczewskitype ideals related to
$\mathbb{FM}$ and $\mathbb{IE}$, proving some results of orthogonality
with other tree forcings ideals.
[1] A. Roslanowski, On game ideals, Colloq. Math., 59(2):159168, 1990.
[2] O. Spinas, Perfect set theorems, Fund. Math., 201(2):179195, 2008.
 Javier Legris.
Existential Graphs as a Framework for Introducing Logical Constants and Logical Systems
Abstract: Charles
S. Peirce conceived his diagrammatic logic of the Existential Graphs
(EGs) as the best formulation of logical systems, according to his
iconic conception of logic and his idea of “exact logic”. Consequently
they constituted a general methodology for the mathematical treatment of
deduction based on notions from topology. Thus, the EGs open a fully
new direction in symbolic logic. Peirce formulated different
diagrammatic systems for the EGs that have been intensively studied and
expanded in the last years.
In this paper a new interpretation of Peirce’s EGs is introduced by
applying them as a framework for defining logical constants and logical
systems. This interpretation is coherent with the idea of the
‘productive ambiguity’ of diagrammatic representation implied in
Peirce’s own conception. Diagrams are open to different interpretations
because of their purely structural nature. Furthermore, this
interpretation attempts to build up a bridge between Peirce’s EGs and
the current perspective in logic of Structural Reasoning. According to
this, EGs could be properly understood as a kind of Structural
Reasoning, in the sense used by Peter SchroederHeister and originated
in Gerhard Gentzen’s ideas. Instead of using sequent style systems, the
EGs can be used in order to formulate general properties of deduction
and to define logical concepts. Instead of combinatorial analysis and
recursion (as in the case of sequent systems), EGs can be studied by
topology. In this sense, the EGs can be interpreted as an account of
logical structures. More specifically, the ‘scroll’ formulated by Peirce
in early formulations of the EGs will be used to express an
implication structure.
The main claims of the paper can be summarized as follows: (1)
diagrams can be used in two different senses: (a) as an intuitive way to
express the nature of deduction in terms of diagrams, (b) as a
mathematical tool to characterize logical constants of different logic
systems. (2) The theory for formulating logic systems evolves from a
coherent philosophical perspective, so that there is a clear
correspondence between logical theory (including a topological
framework) and philosophical theory (the iconic nature of deduction).
(3) The philosophical theory does not make deduction and logical
concepts dependent on ordinary language.
 Philipp Lücke.
Infinite fields with free automorphism groups
Abstract: Shelah
proved that a free group of uncountable rank is not isomorphic to the
automorphism group of a countable firstorder structure. In contrast,
Just, Shelah and Thomas showed that it is consistent with the axioms of
set theory that there is a field of cardinality $\aleph_1$ whose
automorphism group is a free group of rank $2^{\aleph_1}$. Motivated by
this result, they ask whether there always is a field of cardinality
$\aleph_1$ whose automorphism group is a free group of rank greater than
$\aleph_1$.
I will present joint work with Saharon Shelah that shows that the
free group of rank $2^\kappa$ is isomorphic to the automorphism group of
a field of cardinality $\kappa$ whenever $\kappa$ is a cardinal
satisfying $\kappa=\kappa^{\aleph_0}$. The techniques developed in the
proof of this statement also show that the existence of a cardinal
$\kappa$ of uncountable cofinality with the property that there is no
field of cardinality $\kappa$ whose automorphism group is a free group
of rank greater than $\kappa$ implies the existence of large cardinals
in certain inner models.
 Roman Murawski.
Philosophy of logic and mathematics in Warsaw School (the case of Tarski and Mostowski)
Abstract: The aim of the talk is to analyze main trends and tendencies, main standpoints and
views in the philosophical reflection on mathematics and logic in Warsaw School between
the wars.
This was the time of intensive development of mathematics (Polish Mathematical
School) and of logic (Warsaw Logical School, LvovWarsaw Philosophical School) in
Poland. Hence a natural question arises whether this development of mathematics and
logic was accompanied by philosophical reflection on those disciplines, whether the researches
were founded on and stimulated by certain fixed philosophical presuppositions.
On the other hand philosophy of mathematics and logic is based on and uses certain
results of metamathematics, of the foundations of mathematics and of logic. Did logical
achievements influence the philosophical reflection?
The philosophical views of main logicians and mathematician (among them of Tarski
and Mostowski) will be presented and analyzed.
We come to the conclusion that Polish logicians and mathematicians being convinced
of the importance of philosophical problems and knowing quite well the current philosophical
trends treated logic and mathematics as autonomous disciplines independent
of philosophical reflection on them, independent of any philosophical presuppositions.
Therefore they sharply separated mathematical and logical research practice and philosophical
discussions concerning logic and mathematics. Philosophical views and opinions
were treated as “private” matter that should not influence the mathematical and metamathematical
investigations. On the contrary, in the latter all correct methods could and
should be used. This “methodological Platonism” enabled Polish logicians and mathematicians
to work in various areas without being preoccupied by philosophical dogmas.
In controversial cases, as for example in the case of the axiom of choice in set theory, their
attitude can be characterized as neutral – without making any philosophical declarations
they simply considered and studied various mathematical consequences of both accepting
and rejecting the controversial principles and investigated their role in mathematics.
 Hugo Nobrega.
Obtaining Weihrauchcomplete functions and relations from sets of real numbers
Abstract: We report on research that connects the theory of Weihrauch degrees to the descriptive set theory of classes of functions.
Among other previously known results, functions $f_n$ for each
natural number $n$ and a relation $R$ had already been found which
characterize the Baire class $n$ functions and the class of functions
preserving $\mathbf{\Sigma}^0_2$ under preimages, respectively, in the
sense that a given function is in one of the mentioned classes if and
only if it is Weihrauchreducible to the corresponding function or
relation.
In this talk, we show that functions characterizing the Baire class
$\alpha$ functions can be obtained by a general construction applied to
any Wadgecomplete set for $\mathbf{\Sigma}^0_\alpha$, for all $\alpha
< \omega_1$, and show that the $f_n$ functions are special cases of
this construction.
Furthermore, we show that relations characterizing the classes of
functions with a certain type of partition property related to the
famous JayneRogers theorem  namely, for each $\alpha < \omega_1$,
the class of functions whose domain can be partitioned into countably
many $\mathbf{\Pi}^0_\alpha$ sets such that the function restricted to
each set in the partition is continuous  can also be obtained by
another general construction applied to any Wadgecomplete set for
$\mathbf{\Sigma}^0_\alpha$.
In particular, using the JayneRogers theorem we show that $R$ can also be seen as a special case of this construction.
 Edoardo Rivello. Revision without Choice
Abstract: Revision
sequences were introduced in 1982 by Herzberger and Gupta
(independently) as a mathematical tool in developing what is since then
known as "The revision theory of Truth". Later, the same tool was
applied to other areas of logic and philosophy, like the theory of
definitions, the analysis of circular concepts, the theory of rational
belief, and others. In the original and informal presentation of the
theory, the revision sequences are ordinallength sequences, namely
operations defined on the proper class of all ordinal numbers. However,
it is a common feature of most revision theories proposed in the
literature the fact that the same theory can be obtained by quantifying
over the class of all limitlength revision sequences, so providing a
formalisation of the theory in ZFC. Further, for a given theory, by an
application of the LowenheimSkolem theorem, McGee’s theorem establishes
an upper bound for the length of the revision sequences we need to
quantify over.
McGee's theorem and analogous or related results, are usually proved
by invoking some amount of the Axiom of Choice, in the form of the
LowenheimSkolem theorem or in another. In this talk I will present a
general method for reproving such results without using any form of the
Axiom of Choice. The method applies to a wide range of revision theories
encompassing Harzberger's, Gupta's and Belnap's original ones.
The talk is based on the following recent works (preprints available on academia.edu):
E. Rivello, StabilityDependent Revision, submitted, 2013.
E. Rivello, Revision without the Axiom of Choice, submitted, 2014.
 Carsten Rösnick.
On Type2 Complexity of Some Operators in Geometry, Topology and Analysis
Abstract: This
talk will be about my yettobefinished PhD thesis. In my thesis I
analyze the computational complexity of some operators in geometry and
analysis based on earlier work of Ko and Friedman (1981/84, 1991), Chou
and Ko (1995, 2005), Brattka et al (1999, 2003), Labhalla et al (2001),
Zhao and Müller (2008), Kawamura and Cook (2011), and others. Examples
of operators include binary set union, intersection and projection, as
well as integration, differentiation, reciprocals $f \mapsto 1/f$ and
inverses $f \mapsto f^{1}$. These, and more, operators are realized (in
the TTEsense/"Weihrauch model") on oracle Turing machines where their
realizers are type2 functionals, i.e. functionals $\phi$ of form
$\mathbb{N}^{\mathbb{N}} \times \mathbb{N} \to \mathbb{N}$.
In order to speak about
computations, we first have to introduce representations for spaces like
the class of all closed subsets of $\mathbb{R}^n$, or the class of continuous,
smooth or analytic functions. Usually, there is more than one "natural"
choice of representation, but as it will turn out, most of the ones we
will see are polytime equivalentthus, from the perspective of
complexity, it does not matter which one we choose when devising the
complexity of operators. This in a sense even proves the complexity
bounds for operators to be 'robust'. However, obviously not all
representations turn out to be polytime equivalent: For example, there
are three quite natural representations in the realm of closed sets
which are not polytime equivalent; the first two because of the
difficulty to get global information out of local information, the
second two due to the need to add additional data.
Having settled
the question about some representations and their relations, we turn to
operators: the projection for sets, as well as integration and function
inversion for continuous functions. Projection proves to be polytime if
restricted to convex compact sets; while the lack of convexity links it
to $P$ vs $NP$ in the discrete world. Integration is known to map polytime
functions to polytime ones when restricted to analytic functions, but
over smooth functions it is not polytimejust like for continuous
functions. So, in a sense, smoothness does not help from a complexity
point of view. This gap between polytime over analytic and exptime over
smooth functions can be quantified in terms of the Gevrey level
(loosening the bounds on the growth of analytic functions depending on
said level)which we will do. It turns out that the complexity of
integration depends exponentially on this level. We will also take a
look at a third operator, function inversion $f \mapsto f^{1}$,
sufficient preconditions for this operator to be at least computable,
and the barriers to surpass due to its linkage to the existence of one
way functions and one way permutations.
 Sam Sanders.
Computing with infinitesimals
Abstract: See PDF document.
 Ralf Schindler, Sandra Uhlenbrock and W. Hugh Woodin.
Producing Iterable Models with Woodin Cardinals from Optimal Determinacy Hypotheses
Abstract: Projective
determinacy is the statement that for certain infinite games, where the
winning condition is projective, there is always a winning strategy for
one of the two players. It has many nice consequences which are not
decided by ZFC alone, e.g. that every projective set of reals is
Lebesgue measurable. An old so far unpublished result by W. Hugh Woodin
is that one can derive specific countable iterable models with Woodin
cardinals from this assumption. Work by Itay Neeman shows the converse
direction, i.e. projective determinacy is in fact equivalent to the
existence of such models. These results connect the areas of inner model
theory and descriptive set theory. We will give an overview of the
relevant topics in both fields and, if time allows, sketch a proof of
Woodin's result.
 Peter Schuster and Davide Rinaldi.
Eliminating Krull's Lemma for Horn Clauses
Abstract: In
the language of rings we consider implications from a finite
conjunction to a finite disjunction of atomic formulas. It is wellknown
that one can characterise algebraically when such implication can be
proved for (commutative) rings and for integral domains, respectively.
By interpolation, we extend this characterisation to the provability of
such implications for reduced rings: i.e., rings in which only 0 is
nilpotent. Scott entailment relations are particulary suited for proving
those characterisations. From the monoconclusion case one can
read off a syntactical proof that the theory of integral domains is
conservative for Horn clauses over the theory of reduced rings. This
allows to eliminate Krull's Lemma, the appropriate incarnation of Zorn's
Lemma, from many a "short and elegant'' proof in commutative algebra
that works by reduction modulo a generic prime ideal. With only
little effort, the approach above can be carried over from the case of
integral domains over reduced rings to the case of local rings over
rings: just replace ideals by filters, and "is zero'' by "is
invertible''. With hardly more effort one can also deal with occurrences
of mixed predicates in the antecedent. All proofs on the
metalevel are constructive and elementary. Within the given theories,
however, it is irrelevant whether one thinks provability in terms of
classical or intuitionistic logic: all formulas under consideration are
geometric implications. In competition with related results from
dynamical algebra, we eliminate also the condition that every integral
domain or local ring be nontrivial, and thus avoid to speak of
absurdity. This opens up the possibility to step down from classical
logic not only to intuitionistic but also to minimal logic.
 Johannes Stern.
A New Norm for Truth and the Modal Logics of Axiomatic Theories of Truth
Abstract: Due to a seminal result by Solovay it is well known that the modal
logic $GL$ is the modal logic of the natural provability predicate of $PA$
in $PA$. In other words $GL$ expresses exactly the modal properties of the
natural provability predicate of $PA$. At least in principle one could try
determining the modal logic of other sentential predicates of an
arithmetic theory or an extension thereof. In particular it seems
interesting to look for the modal logic of the truth predicate of
various axiomatic theories of truth. While this question is interesting
in its own right we shall focus on its philosophical relevance and argue
that such an investigation leads to a new norm for theories of
reflexive truth.
We argue for this this claim by discussing the recent arithmetic
completeness result by Czarniecki and Zdanowski. Czarnecki
and Zdanowski show that the normal modal logic $KDD_C$, which extends
the basic normal modal logic $K$ by the modal principles
$$\begin{align*}
(D)&&\Box\phi\rightarrow\Diamond\phi&&\\
(D_C)&&\Diamond\phi\rightarrow\Box\phi&&
\end{align*}$$
is the modal logic of the truth predicate of symmetric theories of
truth (although Czarnecki and Zdanowski do not mention this fact). A
theory of truth is called symmetric if and only if the truth predicate
commutes with all boolean connectives and is closed under the rules of
Tintroduction and Telimination. The ingenious strategy employed by
Czarnecki and Zdanowski in showing the arithmetic completeness of
$KDD_C$ with respect to symmetric theories of truth is to show that
$KDD_C$ is the maximal normal modal logic that can be consistently
extended by the relevant modal instances of the diagonal lemma. Since
$KDD_C$ is sound with respect to symmetric theories of truth it then
must be the modal logic of the truth predicate of symmetric theories of
truth.
The observation that $KDD_C$ is a maximal modal logic lead us to
proposing the following new norm for reflexive theories of truth:
The modal logic of the truth predicate of reflexive
theories of truth ought to be maximal (in the sense of Czarnecki and
Zdanowski).
We argue that this norm should replace the naive desiderata that all
Tbiconditionals ought to be derivable, which, however, cannot be
satisfied in classical and consistent theories of truth. We then
illustrate and discuss the consequences of adopting such a norm and also
point to certain limitations.
 Elena Tatievskaya.
Symbol and Rule: Cassirer’s treatment of Kant’s Schematism
Abstract: Cassirer
(The Philosophy of Symbolic Forms (19231929)) regards cognition as
symbolizing that follows rules. Such rules are embodied in systems of
meaning invariants used to order objects and govern scientific as well
as prescientific cognition, language and art. Cassirer compares symbols
with Kant’s transcendental schemata that mediate between categories and
sensible appearances and uses the terms “schema” and “rule” as
interchangeable. Hence it may be asked what the rulelike function of
the schema consists in and whether this notion therefore could help to
clarify Cassirer’s concept of a symbol and of how rules of symbolizing
work.
I argue that transcendental schemata can be regarded as belonging to
the rules of understanding that allow of judging. Using Krausser’s
definition of the topological structure of the intuition of time
(„Kant’s Schematism of the Categories and the Problem of Pattern
Recognition“, Synthese 33, 1976, 175192) which presupposes the
relations of simultaneity and following between different elements of a
sensibly intuited manifold I propose to represent principles of pure
understanding and socalled “schemajudgments” or “criteria” that
connect schemata with categories in such a way that the principles
define possible antecedents of the propositions that fix such criteria.
The consequents of such propositions involve the expression of the form
of the judgment that becomes possible due to the application of the
schema. Thus a transcendental schema works as a rule according to which
the category represented by the schema defines some logical property of
the judgment that concerns the apprehended object. Such a property could
be the quantity, quality, form or modality of the judgment.
If we take for example the categories of quantity and consider the
number as “Anzahl” of the elements of a sensibly given manifold of which
some property is predicated in a judgment the general principle of the
axioms of intuition that requires that all intuitions be extensive
quantities can be formulated in the form “Either an appearance differs
from other appearances simultaneous with it insofar as they in their
turn are not simultaneous with each other or some appearances of the
given manifold differ with respect to their simultaneity from the other
ones or the given manifold contains no appearances that are not
simultaneous with at least one of its elements”. Then the proposition
that represents the schema of e.g. the category of plurality is “If a
given manifold contains an appearance that is simultaneous with only
some of the other appearances and differs with respect to simultaneity
from the remaining ones a certain property can be predicated of some of
the appearances of the manifold”.
Cassirer’s symbol functions analogously to a schema representing a
category since it orders objects due to its ability to represent a
system of meaning invariants which in its turn determines the rulelike
role of the symbol. The cognition is not restricted to the formation of
judgments but involves the production of symbols of different kinds.
 Thilo Weinert.
On Partitioning Linearly Ordered Quadruples in Canonical NonChoiceContexts
Abstract: See PDF document
 Gregory Wheeler.
Beware of B
Abstract: No
epistemic logic should include the Brouwersche schema, (B), as (B) and
(K) yield a pair of known schemas (Chellas 1980) which I dub the BUM
schemas. BUMs are a disaster for epistemic logics in general, although
BUMs are not problematic for provability logics, which suggests that (B)
points out a sharp distinction between (regular) epistemic logics and
logics of provability. To mark the distinction, and to preserve
Brouwer’s good name, I propose epistemic logicians hereafter refer to
(B) as the Blog Schema. Finally, with the downfall of (B) joining the
ranks of (5) (Hintikka 1962) and (4) (Williamson 2000), hints of a new
case against regular systems of epistemic logic comes into view: (K)
does not play well with other modal schemas.
 Gunnar Wilken and Andreas Weiermann.
Goodstein sequences for prominent ordinals up to the ordinal of $\Pi^1_1CA_0$
Abstract: We present a smooth approach to generate Goodstein
principles which are true but unprovable in theories of iterated
inductive definitions, $ID_n$, and eventually the theory $\Pi^1_1CA_0$.
This work was published in the Annals of Pure and Applied Logic in 2013.
