Fall 2025, Set Theory and Mathematical Reasoning

There are two main goals for this class:

• Develop some skills of mathematical proof (both written and verbal)

• Learn some basic set theory (both informal and formal)

You will learn both of these things by doing them.

Every week, you will write up the proof of one result we have gone over in class, and turn in the written version on Gradescope. I will give many choices each week of which result to write up - I encourage you to choose one that will provide an appropriate challenge for you, whether that's getting precise on concisely writing up a simple proof, or managing the organization of a more complex proof.

Roughly half of the class time on every day after the first will consist of student presentations of mathematical results. Everyone should sign up to do one of these presentations in Part I and one of these presentations in Part II. I'll be happy to meet with you in the days leading up to your presentations to ensure that you've figured out how your result works, and give some feedback on effectively presenting it.

Your final grade will primarily be based on completing all of these written and in-class proofs, with only slight modifications for quality.

Part I: Informal set theory

Week 0 (9/25): Sets as meanings of words

Week 1 (9/30, 10/2): Sets and numbers

Presenters: Ava

Week 2 (10/7, 10/9): Well-orderings, countable vs uncountable infinities

Presenters: Landon, ?, ?, ?, ?, ?

Week 3 (10/14, 10/16): Many countable infinities

Presenters: Kailey, ?, ?, ?, ?, ?

Week 4 (10/21, 10/23): The real numbers, and the power set of the natural numbers

Presenters: Frankie,?, ?, ?, ?, ? 

Week 5 (10/28, 10/30): The Axiom of Choice, the paradoxes

Presenters: ?, ?, ?, ?, ?, ?

Part II: Formal axiomatic set theory

Week 6 (11/4, 11/6): The formal language, and the "small" axioms

Presenters: Frankie, 

Week 7 (11/11, 11/13): Power set, separation, and replacement

Presenters:

Week 8 (11/18, 11/20): Binary relations, well-orderings, and the axiom of infinity

Presenters:

Week 9 (11/25, Thanksgiving): Formalized ordinals, and the well-ordering principle

Presenters:

Week 10 (12/2, 12/4): The Continuum Hypothesis and further topics

Presenters:

• Videos about proof by induction

  • Khan Academy, straightforward video showing how to prove that the sum of the first n natural numbers is n(n+1)/2

  • Kimberly Brehm, slightly longer video beginning with the general theory of mathematical induction (and why it works for integers, but not rationals or real numbers), and then showing how to use it to prove that the sum of the first n odd numbers is n squared

    • Her video about strong induction, where the induction step involves assuming the result is true all the way up to the point you are extending

  • Trefor Bazett, another video showing the same result as the Khan Academy one, with a general "ladder" metaphor for understanding induction

    • His video about strong induction

  • Zach Star, What Does Mathematical Induction Really Look Like?, shows visually how induction works for various geometric examples (including the Towers of Hanoi puzzle, and dominoes)

  • EH, Introduction to Strong Induction, gives a helpful example proof where strong induction is essential

  • AfterMath, Induction, Weak and Strong - one video with both concepts, explaining

• Proofs without words

  • Examples from Arindam Khan, on Quora

  • Robin Miller, "On Proofs Without Words", 2012 - discussion of what it takes for a diagram to constitute a proof without words

• Work by Fenner Tanswell on proofs (some of his other papers, and his book, may also be of interest):

  • Keith Weber and Fenner Tanswell, "Instructions and recipes in mathematical proofs", 2022 - points out that mathematical proofs, like recipes for cooking, are usually written in the imperative mood, telling the reader what to do, see, or notice

  • Fenner Tanswell and Matthew Inglis, "The Language of Proofs: A Philosophical Corpus Linguistics Study of Instructions and Imperatives in Mathematical Text", 2023 - does a more detailed study of the specific verbs used in mathematics, particularly noticing the verbs “let”, “consider”, “assume”, “denote”, “note”, “define”, “suppose”, “recall”, “write”, “take”, “choose”, “fix”, and “observe”.

• My work on proofs:

  • Kenny Easwaran, "Probabilistic Proofs and Transferability", 2009 - argues that the point of a mathematical proof is not to get the reader to trust the author, but to give the reader a way to produce their own knowledge of the result

  • Kenny Easwaran, "Rebutting and Undercutting in Mathematics", 2015 - argues that although published mathematical proofs are not infallible, they are expected to reach a level of detail at which a counterexample would show which step went wrong

• Yehuda Rav, "Why Do We Prove Theorems?", 1999 - a classic paper arguing that the point of proofs is not just to provide knowledge of the result but to develop methods and techniques for other applications

• Lewis Carroll, "What the Tortoise Said to Achilles", 1895 - a classic paper making the case that what is important for understanding a proof is not a knowledge of the statement of an argument, but an ability to follow the argument

• Trefor Bazett's discrete math videos

• Antonio Montalban's set theory class videos

• José Ferreirós, The Early Development of Set Theory, at the Stanford Encyclopedia of Philosophy