- Wednesday 13/01/2021, 17:00 hrs, Week 1,
Zoom 968 6020 1432.

**Logic Day Special**.*Insights into Participant's Logic Research*.

This week's Logic Seminar on Wed 13 January 2021 at 17:00 hrs is an open session where, in light of the World Logic Day on Thursday, everyone is encouraged to give a 5 to 10 minutes presentation about his favourate result or results of his own work.

PS: In this session, several speakers from the audience and the organisors shared important or recent results. Two of them gave their slides for the public: Andre Nies; Frank Stephan.

- Wednesday 20/01/2021, 16:00 hrs, Week 2,
**Brian Rabern**.*Against Fregian Quantification*.

Talk given at FASS moderated by Ben Blumson, NUS. Please see at FASS for further informations; the Logic seminar itself has no speaker and the organisors believe that this talk about quantification in mathematics by Frege and Tarski will meet the general interest of mathematicians.

- Wednesday 27/01/2021, 17:00 hrs, Week 3,
Zoom 968 6020 1432.

**No Seminar.**

- Wednesday 03/02/2021, 17:00 hrs, Week 4,
Zoom 968 6020 1432.

**Wong Tin Lok.***Arithmetic under negated induction*.

Arithmetic generally does not admit any non-trivial quantifier elimination. I will talk about one exception, where the negation of an induction axiom is included in the theory. Here the Weak König's Lemma from reverse mathematics arises as a model completion.

This work is joint with Marta Fiori-Carones, Leszek Aleksander Kolodziejczyk and Keita Yokoyama.

- Wednesday 10/02/2021, 17:00 hrs, Week 5,
Zoom 968 6020 1432.

**No Seminar**.

- Wednesday 17/02/2021, 17:00 hrs, Week 6,
Zoom 968 6020 1432.

**Xiao Ming**.*Borel Order Dimensions*.

Order dimension is a classical combinatorial object and has been widely studied by set theorists, combinatorists and computer scientists since its introduction by Dushnik and Miller in 1941. We focus on the partial orderings that are definable as a Borel subsets in a Polish space and analyze the order dimension that can be realized by Borel definable orders and show that there are some interesting behaviors that can be quite different from the classical order dimension using arbitrary realization.

This is a joint work with Dilip Raghavan.

- Wednesday 03/03/2021, 17:00 hrs, Week 7,
Zoom 968 6020 1432.

**Desmond Lau**.*On the unification of two maximal axioms*.

Martin's Maximum^{++}and Woodin's axiom (*) are two statements independent of, but consistent with, ZFC. I will present the common reasons they are appealing as set-theoretic axioms, before comparing the sense in which they are maximal. I will also run through an exposition of the recent work by Asperó and Schindler, which shows Martin's Maximum^{++}implies (*), effectively unifying the statements.

- Wednesday 10/03/2021, 17:00 hrs, Week 8,
Zoom 968 6020 1432.

**Zekun Jia**.*Two Ramsey-theoretic statements in models where AC fails*.

There are a lot of theorems in Ramsey theory whose proof explicitly or implicitly uses the Axiom of Choice. This talk will focus on Ramsey's Theorem and Open Ramsey Theorem in three models of set theory where the Axiom of Choice fails (the basic Cohen model, the basic Fraenkel model, and the ordered Mostowski model), as well as some consistency and independence results that follow. Also, the usual proof of Open Ramsey Theorem on omega given by Galvin and Prikry assumes the Axiom of Dependent Choice, and this talk will sketch an improvement on that proof to make it purely constructive.

This talk is about a project advised by Zach Norwood.

- Thursday 18/03/2021, 10:00 hrs (17/03/2021
in USA), Week 9,
Zoom 968 6020 1432.

**Ko Liling**.*Towards finding a lattice of fickleness strictly above*.**ω**^{2}

Given a finite lattice**L**that can be embedded in the recursively enumerable (r.e.) Turing degrees**(R,≤)**we do not in general know how to characterize the degrees**d ∈ R**below which**L**can be bounded. The important characterizations known so far are of the**L**and 1-3-1 lattices, where the former is bounded exactly by the degrees with fickleness strictly above_{7}**ω**and the latter is bounded exactly by the degrees containing sets of fickleness greater equal**ω**. Given that the fickleness hierarchy collapses exactly to the powers of^{ω}**ω**with the first few levels being**0,ω,ω**we want to find a lattice that characterizes the levels strictly above^{2},...,ω^{ω}**ω**. We begin by exhausting the lattices^{2}**L**that are as ``small'' as**L**and 1-3-1, but these lattices turn out to characterize the levels strictly above_{7}**ω**or from**ω**onwards, if^{ω}**L**is not already embeddable below all non-zero r.e. degrees. We even considered small infinite lattices but they too behave like**L**or 1-3-1. We discovered three lattices besides 1-3-1 that also characterize the levels from_{7}**ω**onwards. Our search for a candidate characterising the levels strictly above^{ω}**ω**therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four sublattices which characterise the levels from^{2}**ω**onwards as a sublattice. Using this criterion as a heuristic we introduce the wide diamond lattice as a candidate, though we conjecture that this lattice also behaves like 1-3-1.^{ω}

- Wednesday 24/03/2021, 17:00 hrs, Week 10,
Zoom 968 6020 1432.

**Philipp Schlicht**.*Sets and graphs in generalised descriptive set theory*.

While descriptive set theory studies definable sets of words of length omega, generalised descriptive set theory studies words of uncountable regular length. The talk will begin with an introduction to this field and its applications. I will then talk about how one can characterise when a definable set is small or admits a colouring with few colours with respect to an open graph.

This is joint work with Dorottya Sziraki.

- Wednesday 31/03/2021, 17:00 hrs, Week 11,
Zoom 968 6020 1432.

**Lars Kristiansen**.*Classic representations of irrational numbers seen from a computability and complexity-theoretic perspective.*

We will address the following question:

Do we need, or do we not need, unbounded search in order to convert one representation of an irrational number into another representation?

We will consider well known representations like Cauchy sequences, Dedekind cuts, base-2 expansions, base-10 expansions and continued fractions, and maybe a few less well-know representations.

A hand-out is available.

- Wednesday 07/04/2021, 17:00 hrs, Week 12,
Zoom 968 6020 1432.

**Frank Stephan**.*On Trees without Hyperimmune Branches*.

In the year 2013, Keng Meng Ng, Frank Stephan, Yue Yang and Liang Yu published the paper "Computational Aspects of the Hyperimmune-free Degrees" and one of the results was given without a full proof. The current talk gives the full proof of this result. In particular, the talk provides the full details of the construction of an co-r.e. infinite binary tree with uncountablly many branches which are all hyperimmune-free, Schnorr-trivial, jump traceable, generalised low and of minimal Turing degree. Hyperimmune-free means that every function Turing reducible to it is majorised by a recursive function. Jump traceable means that for every e one can compute an explicit bound on the number of elements which some further set also depending on e enumerates and that finite set contains the Jump value at e. Generalised low means that the jump of A is recursive in the join of A and K. A minimal Turing degree is a nonrecursive Turing degree below which is only the recursive one. Schnorr trivial means that for every f truth-table reducible to the set there is a recursive function which lists out for each input x a set of x+1 many values with one of them being f(x).

The slides of the quite technical talk are here: ps-file, pdf-file.

- Wednesday 14/04/2021, 17:00 hrs, Week 13,
Zoom 968 6020 1432.

**Karen Seidel**.*Learning from informant*.

Learning from positive and negative information, so called informant, is one of the models for human and machine learning introduced by Gold. We review existing classical and recent results regarding the learning power of associated settings.