Infinity — and beyond?

Andrew Brooke-Taylor

Engineering and Physical Sciences Research Council

Early Career Fellow

School of Mathematics

University of Bristol

June 29, 2015

Potential or actual infinity?

ISometimes “infinity” is just convenient terminology. . .

IE.g. “goes to infinity” usually means “grows without bound”.

1, 2, 3, 4, 5, 6, 7, . . .

I. . . but sometimes we really want to talk about an infinitude

all at once.

IE.g. the set of all natural numbers, N = {1, 2, 3, 4, . . .}

Potential or actual infinity?

ISometimes “infinity” is just convenient terminology. . .

IE.g. “goes to infinity” usually means “grows without bound”.

1, 2, 3, 4, 5, 6, 7, . . .

I. . . but sometimes we really want to talk about an infinitude

all at once.

IE.g. the set of all natural numbers, N = {1, 2, 3, 4, . . .}

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

Zeno’s paradox: Achilles and the tortoise

Zeno’s conclusion: Achilles never catches the tortoise.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start.

So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

0 100

200150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start.

So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

0 seconds

0 100

200150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start. So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

0 seconds

0 100 200

150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start. So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

10 seconds

0 100 200150

175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start. So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

15 seconds

0 100 200150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start. So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

17.5 seconds

0 100 200150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Say Achilles runs at 10m/s, the tortoise half that speed, and the

tortoise has a 100m head start. So Achilles will catch the tortoise

after running 200m, i.e., after 20 seconds.

17.5 seconds

0 100 200150 175

Zeno’s infinitely many stages all happen before the 20 second mark.

What’s going on?

Two notions of size:

I length (distance/duration)

I number (of points in a set)

Just because you’re infinite in the second sense, doesn’t mean

you’re infinite in the first sense.

What’s going on?

Two notions of size:

I length (distance/duration)

I number (of points in a set)

Just because you’re infinite in the second sense, doesn’t mean

you’re infinite in the first sense.

What’s going on?

Two notions of size:

I length (distance/duration)

I number (of points in a set)

Just because you’re infinite in the second sense, doesn’t mean

you’re infinite in the first sense.

What’s going on?

Two notions of size:

I length (distance/duration)

I number (of points in a set)

Just because you’re infinite in the second sense, doesn’t mean

you’re infinite in the first sense.

What’s going on?

Two notions of size:

I length (distance/duration)

I number (of points in a set)

Just because you’re infinite in the second sense, doesn’t mean

you’re infinite in the first sense.

An important idea

The limit of an infinite sequence of ever-improving approximations

is the precise value.

0,1

2,3

4,7

8, . . . converges to 1

0.9, 0.99, 0.999, 0.9999, . . . converges to 1

Viewing 0.999 . . . as the limit of the second sequence, we see that0.999 . . . = 1.

An important idea

The limit of an infinite sequence of ever-improving approximations

is the precise value.

0,1

2,3

4,7

8, . . . converges to 1

0.9, 0.99, 0.999, 0.9999, . . . converges to 1

Viewing 0.999 . . . as the limit of the second sequence, we see that0.999 . . . = 1.

An important idea

The limit of an infinite sequence of ever-improving approximations

is the precise value.

0,1

2,3

4,7

8, . . . converges to 1

0.9, 0.99, 0.999, 0.9999, . . . converges to 1

Viewing 0.999 . . . as the limit of the second sequence, we see that0.999 . . . = 1.

An important idea

The limit of an infinite sequence of ever-improving approximations

is the precise value.

0,1

2,3

4,7

8, . . . converges to 1

0.9, 0.99, 0.999, 0.9999, . . . converges to 1

Viewing 0.999 . . . as the limit of the second sequence, we see that0.999 . . . = 1.

Where the idea went:

calculus!

Isaac Newton (1643–1727) Gottfried Leibniz (1646-1716)

Where the idea went: calculus!

Isaac Newton (1643–1727) Gottfried Leibniz (1646-1716)

Where the idea went: calculus, made rigorous!

Bernhard Bolzano (1781–1848) Karl Weierstrass (1815–1897) Augustin-Louis Cauchy (1789–1857)

Back to infinity itself

“The same number”

Georg Cantor (1845–1918)

Definition

Two sets have the same cardinality (think: size, in the “number”

sense) if there is a one-to-one correspondence between all of the

elements in the first set and all of the elements in the second set.

Back to infinity itself

“The same number”

Georg Cantor (1845–1918)

Definition

Two sets have the same cardinality (think: size, in the “number”

sense) if there is a one-to-one correspondence between all of the

elements in the first set and all of the elements in the second set.

Back to infinity itself

“The same number”

Georg Cantor (1845–1918)

Definition

Two sets have the same cardinality (think: size, in the “number”

sense) if there is a one-to-one correspondence between all of the

elements in the first set and all of the elements in the second set.

Examples

Odd natural numbers Even natural numbers

1 2

3 4

5 6

7 8

......

Examples

Odd natural numbers Even natural numbers

1 2

3 4

5 6

7 8

......

Examples

Odd natural numbers Even natural numbers

1

+1

// 2

3

+1

// 4

5

+1

// 6

7

+1

// 8

......

Examples

Odd natural numbers Even natural numbers

1+1 // 2

3+1 // 4

5+1 // 6

7+1 // 8

......

Examples

Natural numbers Even natural numbers

1 2

2 4

3 6

4 8

......

Examples

Natural numbers Even natural numbers

1 2

2 4

3 6

4 8

......

Examples

Natural numbers Even natural numbers

1

⇥2

// 2

2

⇥2

// 4

3

⇥2

// 6

4

⇥2

// 8

......

Examples

Natural numbers Even natural numbers

1⇥2 // 2

2⇥2 // 4

3⇥2 // 6

4⇥2 // 8

......

Weird fact ]1

A subset that’s missing some members (a proper subset) can havethe same cardinality as the whole set.

Richard Dedekind (1831–1916)

Dedekind’s definition of infinity: a set is infinite if it has the same

cardinality as a proper subset of itself.

Weird fact ]1

A subset that’s missing some members (a proper subset) can havethe same cardinality as the whole set.

Richard Dedekind (1831–1916)

Dedekind’s definition of infinity: a set is infinite if it has the same

cardinality as a proper subset of itself.

Examples

Natural numbers Integers

1 0

2 1

3 � 1

4 2

5 � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 1

3 � 1

4 2

5 � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 // 1

3 � 1

4 2

5 � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 // 1

3 // � 1

4 2

5 � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 // 1

3 // � 1

4 // 2

5 � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 // 1

3 // � 1

4 // 2

5 // � 2

6 3

......

Examples

Natural numbers Integers

1 // 0

2 // 1

3 // � 1

4 // 2

5 // � 2

6 // 3

......

Weird fact ]2

The natural numbers have the same cardinality as the set of allrational numbers (i.e. all fractions).

Question:

Do all infinite sets have the same cardinality?

Weird fact ]2

The natural numbers have the same cardinality as the set of allrational numbers (i.e. all fractions).

Question:

Do all infinite sets have the same cardinality?

Answer

(Cantor, 1874) No!

In fact, the set of natural numbers has a di↵erent cardinality thanthe set of all real numbers (i.e. points on a line).

Answer

(Cantor, 1874) No!

In fact, the set of natural numbers has a di↵erent cardinality thanthe set of all real numbers (i.e. points on a line).

That’s about the size of it!