Unit · year 1
MU-101 · Foundations & Mathematical Proof
Threads logic · structure7 theorems
How mathematics is built: sets, logic, and the forms of rigorous argument that everything after this relies on.
Theorems in this unit
T-001
The principle of mathematical induction
If a statement holds for 1 and its truth at n forces its truth at n+1, it holds for every natural number.
T-002
The well-ordering principle
Every non-empty set of natural numbers has a least element, and this is equivalent to induction.
T-003
The pigeonhole principle
If n+1 objects are placed in n boxes, some box holds at least two.
T-004
The irrationality of √2
No ratio of integers squares to 2; the proof is the classic argument by contradiction.
T-005
The infinitude of the primes
There is no largest prime; assuming a finite list yields a contradiction.
T-006
The uncountability of the reals
No list can enumerate all real numbers — Cantor's diagonal argument.
T-007
Cantor's theorem
A set never has the same cardinality as its power set: |P(A)| > |A|.