Last 20 minutes of class on Monday, July 27, 2026.
Everyone takes the exam at the same time.
Logic, Foundations of Mathematics, Set Theory, Type Theory. While this exam claims to be a midterm, the questions are multiple choice, short answer, and matching, and do not require any deep thought. It is more like a quiz that assesses knowledge you should have readily available. The exam only counts as a midterm since it reaches back a little in time, with 25% of the exam hitting material you’ve already been quizzed on (logic and math).
You will take the exam on paper. There is a 20 minute time limit.
You may bring three sheets of paper with notes written on both sides of each.
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.”
Logic
Outlined in the preparation for Quiz 2
Mathematics for Computation
Outlined in the preparation for Quiz 3
Set Theory
What is a set?
Why use sets for foundations?
History
Frege
Russell (Russell's Paradox)
Different formalizations
ZFC - the big one
NBG - introduced classes
NF - stratified comprehension
Basic notation
element of (∈)
subset of (⊆)
Axioms of ZFC
Empty set (∅)
Extensionality
Pairing
Union
Power set
Infinity
Foundation
Replacement
Separation
Choice
Encodings of mathematical objects
Natural numbers
generated as in the Axiom of Infinity
Booleans
false is 0
true is 1
The set of all booleans is 2
Ordered pairs
(a,b) = { {a}, {a,b} }
Tuples
(are really nested ordered pairs)
Relations
encoded as sets of ordered pairs
from domain and codomain
subset of powerset of the product of domain and codomain
ambiguity of R² = R o R vs. R² = R x R
Functions
relations with the functional property
Injections
Surjections
Bijections
Sequences
functions with domain the natural numbers
Lists
empty list is the empty set
non-empty list is a pair of an element and a list
Characters
encoded as the code point of the character
Strings
list of characters
Maps
functions
Integers
sets of pairs of natural numbers with the same difference
Rational numbers
set of pairs of integers with the same ratio
denominator must be positive
Real numbers
defined as Dedekind cuts of rational numbers
Cardinals
Finite vs. Infinite
Diagonalization
Countable vs. Uncountable
Cantor’s Theorem
Ordinals
Von Neumann ordinals
Type Theory (Part 1)
What is it
Characteristic notation x:t
Logic emerges from type theory
Inductive definitions
Booleans
Natural numbers
Integers
Rationals
Fixed size numbers
Products (pairs)
Unit
List
Unicode
More custom types
String
Option
Void
Function types (NON-INDUCTIVE)
Notation
Matching syntax
Naming functions
Examples from arithmetic and logic
Composition of functions
Integers
The cond function (important for the exam)
Sugar
List functions
Type inference
Polymorphism
Partial functions
Subtypes
Unions
Intersections
Sums, products, and exponents
Sets
Equality
Not for free in Type Theory
Real numbers
How the formation rules have to be restricted to avoid paradoxes
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!