One of the most mind-expanding sections of 18.090. You learn that the set of natural numbers ( \mathbbN ) and the set of integers ( \mathbbZ ) have the same cardinality (they are countable), but the real numbers ( \mathbbR ) are uncountable (Cantor's diagonal argument).
The defining feature of 18.090 is its total departure from the computation-heavy style of introductory calculus. In a standard calculus class, a problem might ask: Find the derivative of $f(x) = x^2$. The answer is a number or a function.
In 18.090, the questions change entirely. A problem might ask: Prove that the derivative of an even function is an odd function.
"The first few weeks are about unlearning," says one former student. "In calculus, you assume a lot of things are true because the graph looks like it. In IMR, you have to prove the graph actually exists." One of the most mind-expanding sections of 18
The course focuses on the pillars of mathematical logic: set theory, bijections, induction, and the construction of the real numbers. It forces students to grapple with the definition of limits and continuity not as formulas, but as rigorous logical statements involving $\epsilon$ (epsilon) and $\delta$ (delta).
Even though the proofs must be rigorous text, you should draw diagrams to understand what is happening.
Physically split your notebook page. On the left: "Given / Assumptions." On the right: "Goal / Derived Steps." This mimics Fitch-style natural deduction and forces linear clarity. The defining feature of 18
You will master the standard architectures of mathematical proof:
5.1. LaTeX Everywhere
5.2. Voice-to-Proof
5.3. Dark Mode for Theorem-Proving
You begin with truth tables. But MIT does not treat this as trivial. You learn that logical connectives (( \land, \lor, \lnot )) form a Boolean algebra. The key insight here is tautology—statements that are always true regardless of variable values.