**Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 1**Time:** Monday, August 31, 17:00 CEST**Speaker:** Prof. Saharon Shelah, Hebrew University Jerusalem**Title:** Cardinal arithmetic: Cantor’s paradise**Abstract:** We will explain Hilbert’s first problem. Specifically, this asks what the value of the continuum is: Is the number of real numbers equal to aleph1 — the first infinite cardinal above aleph0=the number of natural numbers? Recall that Cantor introduces infinite numbers just as equivalence classes of sets under “there is a bijection”. The problem of the size of the continuum really means: ”What are the laws of cardinal arithmetic, i.e. the arithmetic of infinite numbers”. We will review its history, (including Gödel and Cohen), mention different approaches, explain what is undecidable, and mainly present some positive answers which we have nowadays. Those will be mainly about cofinality arithmetic, the so called pcf theory; but we will also mention cardinal invariants of the continuum.**Information:** Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

** Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 2

**Time:**Tuesday, September 1, 14:15 CEST

**Speaker:**Prof. Saharon Shelah, Hebrew University Jerusalem

**Title:**How large is the continuum?

**Abstract:**Cantor discovered that in mathematics we can distinguish many infinities, called the aleph numbers. The works of Gödel and Cohen told us that we cannot decide what the value of the continuum is, that is, which of the aleph numbers is the answer to the question “How many real numbers are there?”. This still does not stop people from having opinions and arguments. One may like to assume extra axioms which will decide the question (usually as aleph1 or aleph2), and argue that they should and eventually will be adopted. We feel that assuming the continuum is small makes us have equalities which are incidental. So if we can define 10 natural cardinals which are uncountable but at most the continuum, and the continuum is smaller than aleph10, at least two of them will be equal, without any inherent reasons. Such numbers are called cardinal invariants of the continuum, and they arise naturally from various perspectives. We would like to show that they are independent, that is, that there are no non-trivial restrictions on their order. More specifically, we shall try to explain the Cichon diagram and what we cannot tell about it.

**Information:**Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

** Paul Bernays Lectures 2020: Struggling with the size of infinity**, Lecture 3

**Time:**Tuesday, September 1, 16:30 CEST

**Speaker:**Prof. Saharon Shelah, Hebrew University Jerusalem

**Title:**Cardinal invariants of the continuum: are they all independent?

**Abstract:**Experience has shown that in almost all cases, if you define a bunch of cardinal invariants of the continuum, then modulo some easy inequalities, it follows by forcing (the method introduced by Cohen) that there are no more restrictions. Well, those independence results have been mostly for the case of the continuum being at most aleph2. But this seems to be just due to our lack of ability, as the problems are harder. However, this opinion ignores the positive side of having forcing, of us being able to prove independence results: clearing away the rubble of independence results, the cases where we fail may indicate that there are theorems there. We shall on the one hand deal with cases where this succeeds and on the other hand with cofinality arithmetic, and what was not covered in the first lecture.

**Information:**Due to the unusual circumstances of the COVID-19 pandemia, the Paul Bernays Lectures 2020 will take place as a webinar: Link for this webinar. All lectures are given in English and are self-contained. Lecture 1 is aimed at a general audience; lecture 2 and 3 address the scientific community.

**CUNY Logic Seminar (MOPA)****Time:** Wednesday, September 2, 14:00 New York time (20:00 CEST) – note the time**Speaker: **Petr Glivický, Universität Salzburg**Title:** The ω-iterated nonstandard extension of N and Ramsey combinatorics**Abstract:** In the theory of nonstandard methods (traditionally known as nonstandard analysis), each mathematical object (a set) x has a uniquely determined so called nonstandard extension ∗x. In general, ∗x⊋{∗y;y∈x} – that is, besides the original ‘standard’ elements ∗y for y∈x, the set ∗x contains some new ‘nonstandard’ elements.

For instance, some of the nonstandard elements of ∗R can be interpreted as infinitesimals (there is ε∈∗R such that 0<ε<1/n for all n∈N) allowing for nonstandard analysis to be developed in ∗R, while ∗N turns out to be an (at least ℵ1-saturated) nonstandard elementary extension of N (in the language of arithmetic).

While the whole nonstandard real analysis is most naturally developed in ∗R (with just a few advanced topics where using the second extension ∗∗R is convenient, though far from necessary), recent successful applications of nonstandard methods in combinatorics on N have utilized also higher order extensions (n)∗N=∗∗∗⋯∗N with the chain ∗∗∗⋯∗ of length n>2.

In this talk we are going to study the structure of the ω-iterated nonstandard extension ⋅N=⋃n∈ω(n)∗N of N and show how the obtained results shed new light on the complexities of Ramsey combinatorics on N and allow us to drastically simplify proofs of many advanced Ramsey type theorems such as Hindmann’s or Milliken’s and Taylor’s.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.

**CUNY Set Theory Seminar****Time:** Friday, September 4, 14:00 New York time (20:00 CEST)**Speaker:** Mirna Džamonja, IHPST, CNRS-Université Panthéon-Sorbonne Paris**Title:** On logics that make a bridge from the Discrete to the Continuous**Abstract:** We study logics which model the passage between an infinite sequence of finite models to an uncountable limiting object, such as is the case in the context of graphons. Of particular interest is the connection between the countable and the uncountable object that one obtains as the union versus the combinatorial limit of the same sequence.**Information:** The seminar will take place virtually. Please email Victoria Gitman (vgitman@nylogic.org) for the meeting id.