Columbia Undergraduate Math Society

Spring 2021 <<  Summer 2021 Learning Seminar >> Fall 2021

Sundays, 2pm; on Zoom
Topic: Mathematical Logic
Reference: Kenneth Kunen, The Foundations of Mathematics

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Date Speaker Title Abstract
June 6
We will first introduce Summer UMS and then decide on the logistics for the next weeks. We will go over some potential textbooks we could cover and then pick one by vote. Every member will then have the opportunity to sign up to give a talk.
June 13
Anda Tenie
Set Theory is the Theory of Everything
We will begin with an overview of the topics we'll encounter in this seminar: Set Theory, Model Theory, Proof Theory, and Recursion Theory. We will then start discussing set theory, listing the axioms and discussing some particular ones such as Extensionality and Pairing using examples. By the end of the talk, we will get a sense of why all abstract math concepts are in fact set-theoretic.
June 20
David Chen
Building Up from Nothing
We will learn how to use the axioms of ZF set theory (in particular Pairing, Union, and Comprehension) to go from the existence of an abstract set to more familiar and concrete objects, starting with the empty set, as well as discussing when sets cannot exist. Afterwards, we will move on to basic notions in mathematics, including binary relations, functions and well-orderings in the formal context of set theory.
June 27
Zhenfeng Tu
Counting over Natural Numbers
We will introduce the concept of ordinals and arithmetics of ordinals by applying ZF. Later, we will cover induction and recursion on the ordinals. We will conclude with an application of ordinals in measure theory, showing why counting over natural numbers is meaningful.
July 4
Shiyang Shen
From Set Theory to Logic
We will begin with a quick look at the equivalents of the Axiom of Choice (AC) and define cardinal arithmetic (that is, addition, multiplication, and exponentiation) based on it. Then we will look at the Axiom of Foundation to refine the universe into the class of well-founded sets. In the end, we will see how to regard real numbers and finite boolean expressions within the axiomatic set theory.
July 11
Gabriel Ong
Sentences and Strings: An Introduction to Model Theory
We will begin by discussing the motivations behind model theory and draw connections between first order language, natural language, and mathematical structures. Following this, we will briefly examine some aspects of semantics in first-order logical systems. This will give us a strong foundation for further discussions in model theory.
July 18
Cassandra Marcussen
Formal Proof Theory and Connections to Model Theory
We will learn about formal proof theory, with the goal of developing a system of representing proofs as formal objects that is easy to define and analyze. We begin by introducing modus ponens, the simple yet powerful rule of inference in formal proof theory. We then discuss some strategies for constructing proofs, establishing principles that show how informal mathematical arguments can be replicated in the context of formal proof theory. Finally, we connect the syntactic notions of formal proof theory to the semantic notions of model theory and further discuss the foundations of model theory, highlighting the Completeness Theorem and the notion of a complete set of axioms.
July 25
William Dudarov
Elementary Submodels, the Löwenheim-Skolem-Tarski
theorems, and Models of Set Theory
We will extend the foundations of model theory we heard about in the last 2 lectures. Specifically, we will define extensions by definitions, elementary submodels, elementary extensions, and model-completeness. We then prove some basic results about elementary submodels/extensions, specifically showing that a submodel is elementary if it satisfies the Tarski-Vaught criterion, and proving the downward/upward Löwenheim-Skolem-Tarski theorems. Finally, we discuss models of set theory, highlighting transitive models, and formulas that define set-theoretic properties that are absolute across transitive models, in particular.
August 1
Alan Zhao
August 8
Aiden Sagerman
designed by Nilay Kumar, maintained by Anda Tenie