Georg Cantor — Discovering Sets and Infinity

Imagine you are Cantor discovering the ideas, pulled forward by analysis of trigonometric series which lead into a new theory of sets and infinity.

Prologue: Cantor isn’t trying to invent set theory

In the early 1870s, you are not aiming to “found mathematics.” You are trying to understand trigonometric series and, especially, uniqueness:

If a trigonometric series converges to $0$ on an interval (or on “almost all” points), must all its coefficients be $0$ ?

This pushes you into questions like:

Where can a series vanish?
What kinds of exceptional sets of points can exist?

To answer that, you must begin to describe sets of points on the line with surgical precision.

Lesson 1: Point-set anatomy (limit points and derived sets)

Limit points accumulation points

You notice that some sets pile up near certain locations.

$\{1,1/2,1/3,\dots\}$ piles up near $0$ .
$\{1,2,3,\dots\}$ does not pile up anywhere in $\mathbb{R}$ .

So you define:

A point $x$ is a limit point of $A\subset\mathbb{R}$ if every neighborhood of $x$ contains points of $A$ other than $x$ .

Derived set

Collect all limit points:

A' = \{\text{limit points of }A\}.

Examples:

If $A=\{1/n : n\in\mathbb{N}\}$ , then $A' = \{0\}$ .
If $A = \mathbb{Q}\cap[0,1]$ , then $A' = [0,1]$ .

Iterate the operation

You now repeat:

A'' = (A')',\quad A'''=(A'')',\quad \dots

For $A=\{1/n\}$ :

$A' = \{0\}$
$A''=\varnothing$

You are learning to classify sets by what survives repeated “peeling away” of isolated points.

Lesson 2: Perfect sets and a new kind of largeness

A set $P\subset\mathbb{R}$ is perfect if it is closed and has no isolated points. Equivalently (in $\mathbb{R}$ ):

P' = P.

Perfect sets are self-accumulating: every point is a limit point of the set.

This is one of Cantor’s most powerful distinctions:

A set can be topologically rich perfect without containing any interval.

Lesson 3: The Cantor set — dust that is still huge

Construct $C\subset[0,1]$ by repeatedly deleting middle thirds:

Remove $(1/3,2/3)$ .
Remove the middle third of the remaining intervals.
Continue forever.

The remaining set $C$ is the Cantor set.

Key properties (each is a conceptual shock on first encounter):

Closed.
Perfect.
Contains no nontrivial interval.
Yet it is uncountable.

This forces you to separate:

“How many points are there?” from
“Does it contain an interval?” from
“Does it have length?” (measure theory comes later, but the intuition starts here).

Lesson 4: Size by pairing (bijection) — cardinality is born

You decide that two sets have the same “size” if their elements can be paired off perfectly.

Definition (cardinality): $A$ and $B$ have the same size if there exists a bijection $A\leftrightarrow B$ .

Countability

A set is countable if it can be put in bijection with $\mathbb{N}$ .

Important discoveries:

$\mathbb{Z}$ is countable.
$\mathbb{Q}$ is countable (despite being dense).

Lesson 5: Algebraic numbers are countable (1874)

An algebraic number is a root of some integer polynomial:

a_nx^n + a_{n-1}x^{n-1}+\cdots+a_0=0,\quad a_i\in\mathbb{Z},\ a_n\neq 0.

Sketch of the counting idea:

For fixed degree $n$ and coefficient bound $H$ , there are finitely many such polynomials.
Each has finitely many roots.
Countable union of finite sets is countable.

Therefore the algebraic numbers form a countable set.

Lesson 6: Reals are uncountable (1874) — the breakthrough

You ask: can the reals be listed $x_1,x_2,x_3,\dots$ ?

Cantor’s nested-interval argument (analysis-flavored)

Assume $\{x_n\}$ lists $[0,1]$ .

Choose nested closed intervals

I_1 \supset I_2 \supset I_3 \supset \cdots

so that $x_n\notin I_n$ and the lengths shrink to $0$ .

By completeness of $\mathbb{R}$ , the intersection contains exactly one point:

x\in\bigcap_{n=1}^\infty I_n.

But $x\neq x_n$ for all $n$ since $x\in I_n$ while $x_n\notin I_n$ .

Hence $[0,1]$ is uncountable, so $\mathbb{R}$ is uncountable.

The later diagonal argument (1891)

List decimals $x_n = 0.d_{n1}d_{n2}d_{n3}\dots$ . Build

y = 0.c_1c_2c_3\dots

with $c_n\neq d_{nn}$ . Then $y$ differs from $x_n$ in the $n$ th digit, so the list misses $y$ .

Lesson 7: Cantor’s theorem — power sets are always bigger

For any set $A$ :

|A| < |\mathcal{P}(A)|,

where $\mathcal{P}(A)$ is the power set.

Diagonal-style proof: Suppose $f:A\to\mathcal{P}(A)$ is surjective. Define

B=\{a\in A : a\notin f(a)\}.

If $f(b)=B$ , ask whether $b\in B$ .

If $b\in B$ , then $b\notin f(b)=B$ .
If $b\notin B$ , then $b\in f(b)=B$ .

Contradiction either way. So no surjection exists, hence the strict inequality.

Consequence: There is no “largest” infinity. From any infinity, you can build a strictly bigger one.

Lesson 8: Transfinite numbers — cardinals and ordinals

Cardinals (how many?)

$|\mathbb{N}| = \aleph_0$ .
$|\mathbb{R}| = \mathfrak{c}$ (the continuum).

And

\mathfrak{c} = 2^{\aleph_0} = |\mathcal{P}(\mathbb{N})|.

So you get an endless ladder:

\aleph_0 < 2^{\aleph_0} < 2^{2^{\aleph_0}} < \cdots

Ordinals (what order type?)

To measure well-ordered progressions, you define ordinals.

The first infinite ordinal is $\omega$ (order type of $\mathbb{N}$ ).

Ordinal arithmetic is not commutative:

$1+\omega = \omega$ .
$\omega+1 \neq \omega$ .

This reflects structure, not just size.

Lesson 9: The Continuum Hypothesis (Cantor’s great question)

You know

\aleph_0 < \mathfrak{c}.

Cantor asks: is there a cardinal strictly between them?

Continuum Hypothesis (CH):

\mathfrak{c} = \aleph_1.

(Modern result: CH is independent of ZFC, shown via Gödel and Cohen—far beyond Cantor’s era, but it frames his question.)

Exercises (discovery-aligned)

Derived set practice: Let $A=\{1/n : n\in\mathbb{N}\}\cup\{2\}$ . Find $A'$ and $A''$ .
Countability construction: Give an explicit listing of all pairs $(m,n)\in\mathbb{N}^2$ .
Diagonal method: Assume someone lists all infinite binary sequences. Construct a binary sequence not on their list.
Cantor’s theorem: Reproduce the diagonal subset proof for $|A|<|\mathcal{P}(A)|$ in your own words.