Last 15 minutes of class on Monday, July 20, 2026.
Everyone takes the exam at the same time.
Foundations of Mathematics.
You will take the exam on paper. There is a 15 minute time limit.
You may bring one sheet of paper with notes written on both sides.
Do each of the following to maximize your preparation:
Were you able to check off every box?
The learning objectives for the course up to this point in time include you having developed a familiarity with, and an ability to discuss everything about logic appearing on the course notes.
The fact that active recall is better for acquiring long-term knowledge does not mean that outlines and concept maps are not useful. Learners should use multiple techniques—think “both and” rather than “either or.”
Foundations of Mathematics
Why they are important
Backstory for why they had to develop
Three schools of thought
Logicism
Formalism
Intuitionism
Differences between set theory, type theory, and category theory
Key philosophical differences
Key features of each
Key concepts behind each
Characteristic notations
Applications
Mathematical objects
Booleans
Numbers
Different ways to classify numbers
Operations, including tetration and pentation
Understanding logs
A sense of big numbers
Cardinal and ordinal infinities (WATCH THE VIDEOS)
Pairs
Tuples
Important: these are just nested pairs
Sets
Lots of notation!
Relations
How to write them
Composition ("after")
Reflexive, symmetric, transitive, antisymmetric, asymmetric, irreflexive, etc.
Functions
Notation
Naming
Sugars
Iteration
Composition ("after")
Similarities to relations
Sequences
Functions from natural numbers to the things in the sequence
Lists
Constructed from [] and ::
Not the same as sequences, but similar
Characters
Unit of textual information
Has name and code point
Strings
Just a list of characters, that is the definition
Maps
Similar to functions, but with restricted domain
No real computation to get the value at a key
Others
Graphs, Vectors, Matrices, Tensors, etc.
Encoding of mathematical objects
Basic ideas of set theory encodings
EVERYTHING is a set (wild!)
Basic ideas of type theory encodings
Construction rules
Look like logical arguments (this will turn out to be not a coincidence)
Math vs Computation
Where the two fields diverge
Why Type Theory is a better foundation for computation than Set Theory
This is a mini-quiz which tests for immediate understanding of topics and not your ability to work out problems over an extended duration of time. There is a strict time limit so that your immediate fluency is tested rather than your ability to search the web (or worse, ask a chatbot), since these things take time. There will be 5–10 questions. Some may be multiple choice, multi-select, matching, and very short answer.
All content on the assigned readings is fair game for questions, so do not neglect the readings, and by all means do the recall questions!