These theorems form the bedrock of set theory, governing how we understand the size and structure of sets, both finite and infinite.
Cantor’s Theorem
Statement:
For any set A, the power set ℘(A) (the set of all subsets of A) has strictly greater cardinality than A itself.
Significance:
This proves that there is no “largest” infinite set, establishing an infinite hierarchy of infinities.
Cantor–Bernstein Theorem (Schröder–Bernstein Theorem)
Statement:
If there exists an injective function from set A to set B and an injective function from set B to set A, then there exists a bijection between A and B. Hence, they have the same cardinality.
Significance:
Provides a powerful tool for comparing the sizes of sets without explicitly constructing a one-to-one correspondence.
Zermelo’s Well-Ordering Theorem
Statement:
Every set can be well-ordered; that is, a total order can be imposed such that every non-empty subset has a least element.
Significance:
A cornerstone result equivalent to the Axiom of Choice. It underpins much of modern set theory and transfinite arguments.
Transfinite Induction and Recursion
Statement:
These extend standard mathematical induction to well-ordered sets (such as the class of all ordinal numbers).
Significance:
They allow mathematicians to prove theorems and define objects for all ordinals, making them essential for constructing infinite hierarchies.
Hartogs’ Theorem
Statement:
For any set x, there exists an ordinal that cannot be injected into x.
Significance:
Shows that for any set, there is always a larger well-ordered set, leading to the existence of certain cardinals.
Cantor–Bendixson Theorem
Statement:
Any closed set of real numbers can be uniquely decomposed into:
- A perfect set (a closed set with no isolated points)
- A countable set
Significance:
A classical result in descriptive set theory with deep implications for the structure of sets of real numbers.
Zorn’s Lemma
Statement:
If every chain (totally ordered subset) in a partially ordered set has an upper bound, then the set contains at least one maximal element.
Significance:
Equivalent to the Axiom of Choice and widely used in existence proofs (for example, proving every vector space has a basis).
Banach–Tarski Paradox
Statement:
A solid ball in 3-dimensional space can be decomposed into finitely many disjoint subsets that can be reassembled (using only rotations and translations) into two identical copies of the original ball.
Significance:
A counterintuitive consequence of the Axiom of Choice demonstrating the strange behavior of non-measurable sets.
Famous Problems and Open Questions
The Continuum Hypothesis (CH)
Statement:
There is no set whose cardinality lies strictly between:
- ℵ₀ (countable infinity), and
- 2^ℵ₀ (the continuum).
Status & Significance:
Independent of ZFC. One of the most famous independent statements in mathematics.
Suslin’s Problem
Statement:
Is every complete dense linear order without endpoints that satisfies the countable chain property order-isomorphic to the real line?
Status & Significance:
Independent of ZFC. Led to important combinatorial principles such as ◇ (the Diamond Principle).
The Axiom of Choice (AC)
Statement:
For any collection of non-empty sets, there exists a function choosing one element from each set.
Status & Significance:
Independent of ZF. Powerful but non-constructive. Responsible for many deep results and paradoxical consequences.
The Measure Problem
Statement:
Does there exist a non-trivial measure defined for all subsets of the real numbers?
Status & Significance:
Closely connected to large cardinal axioms. Independent of ZFC.