Part II: Set Theory and the Axiom System

Set theory is the lingua franca of modern mathematics. Almost all mathematical objects can be coded as sets, and almost all mathematical arguments can be formalized in the Zermelo-Fraenkel axiom system.

2.1 Naive Set Theory and Russell's Paradox

Cantor's definition (informal): A set is any collection of definite, well-distinguishable objects of our intuition or thought.

Russell's Paradox (1901): Let R = {x : x ∉ x}. Then R ∈ R ↔ R ∉ R. The naive comprehension principle — for any property P, {x : P(x)} is a set — is inconsistent.

Resolution: Restrict comprehension. Sets may only be formed by separating elements from an already existing set. New sets must be explicitly constructed from existing ones.

2.2 Zermelo-Fraenkel Set Theory (ZFC)

The axioms of ZFC (Zermelo 1908, Fraenkel 1922, with the Axiom of Choice) are the dominant foundation.

Axiom	Statement
Extensionality	Two sets are equal iff they have the same elements
Empty Set	∃∅ with no elements
Pairing	For any a, b, ∃{a,b}
Union	For any family F, ∃⋃F containing all elements of elements of F
Power Set	For any A, ∃𝒫(A) containing all subsets of A
Infinity	There exists an infinite set (the natural numbers can be constructed)
Replacement	If F is a definable function and A is a set, F(A) is a set
Separation	For any set A and property P, {x ∈ A : P(x)} is a set
Foundation	Every non-empty set has an ∈-minimal element (no infinite descending chains)
Choice (C)	For any collection of non-empty sets, there is a function selecting one element from each

What ZFC does: ZFC is strong enough to formalize virtually all of mathematics. Real analysis, algebra, combinatorics, geometry, probability theory — all can be translated into the language of sets and proved from ZFC.

What ZFC cannot do: By Gödel's incompleteness theorems, ZFC cannot prove its own consistency (assuming it is consistent). There are sentences — most famously, the Continuum Hypothesis — that are independent of ZFC: neither provable nor refutable.

2.3 Cardinals and Ordinals

Ordinal numbers encode well-ordering. Every well-ordered set is isomorphic to a unique ordinal. The ordinals are 0, 1, 2, …, ω, ω+1, …, ω·2, …, ω², …, ωω, …, ε₀, …

Cardinal numbers encode size. Two sets have the same cardinality if there is a bijection between them. The cardinal of a set A is denoted |A| or #A.

Cantor's Theorem: For any set A, |A| < |𝒫(A)|. Therefore: ℵ₀ < 2^ℵ₀ < 2^(2^ℵ₀) < … The cardinal hierarchy is strictly increasing and has no maximum.

The Continuum Hypothesis (CH): There is no cardinality strictly between |ℕ| = ℵ₀ and |ℝ| = 2^ℵ₀. Equivalently: 2^ℵ₀ = ℵ₁.

Independence: Gödel (1938) proved CH is consistent with ZFC (it holds in the constructible universe L). Cohen (1963) proved ¬CH is consistent with ZFC (via forcing). Therefore, CH is independent of ZFC. This is a genuine mathematical fact, not a philosophical curiosity: the question of how many reals there are cannot be settled from the standard axioms.

2.4 Alternative Set Theories

NBG (von Neumann-Bernays-Gödel): Adds proper classes (collections too large to be sets, like the class of all sets) without increasing proof-theoretic strength.

NF (Quine's New Foundations): Uses stratified comprehension instead of separation. Allows a universal set. Its consistency relative to ZFC is still not fully established.

IZF / CZF: Intuitionistic set theories that reject the law of excluded middle. Used as foundations for constructive mathematics.

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