Infinity

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, …

1. I’m behind

2. There is always a later time when I’m still behind

3. Drat

Zeno: Surely waiting for

an infinite collection of

times takes an infinite time?

Diogenes: Wait till I catch

you…

12+14+18+116

+⋯+11024

161

1/2 1/2

1/2

1/2

1/2

1/2

1/2

1/4

1/4

1/4

1/4

1/4

1/8

1/8

1/8

1/8

1/4

1/8

161

161

161

161

12+14+18+116

+⋯+11024

¿1−11024

‘so’

tiny thing

same tiny thing

Zeno: Surely adding an infinite collection

of times gives an infinite time

Adding halves: Adding an infinite collection of things (can) give a finite thing

Beermat competition

Beermat stacking rules

Optimal beermat stacking

13

12

141

5

is bigger than

12+2×( 14 )+4×( 18 )+8×( 116 )+…

which is

12+12+12+12+…

which is

How many matsSpan Number of mats

Shaftment (15cm) 6

Cubit (50cm) 227

Yard (90cm) 13000

Ell (115cm) 150000

This bar (10m) 1043

∞

1−1+1−1+1−1+1−…=?

1+(−1+1 )+(−1+1 )+…=1

(1−1 )+(1−1 )+(1−1 )+…=0

12+14+18+… ¿1

¿ ¿12+13+15+… ¿∞

¿ ¿

1−1+1−… ¿¿ ¿

… +

wrathgluttonypride

You just slip out the back, Jack Make a new plan, Stan You don't need to be coy, Roy Just get yourself free Hop on the bus, Gus You don't need to discuss much Just drop off the key, Lee And get yourself free …

1,2,3,4,5,6,7,8,…

size 3 size

ℵ 0

1,2,3,4,5,6,7,8,…

wrath

gluttony

pride

Sets mentioned in my talk

…

1,2,3,4,5,6,7,8,…

wrath

gluttony

pride

Sets mentioned in my talk

…Sets mentioned in my talk

• Sets that don’t contain themselves are ‘plain’.

• Sets that do contain themselves are ‘fancy’.

• The Russell set is the set of all plain sets

• Is the Russell set plain or fancy?

• R is the set of all plain sets.• If R is a plain set, it is in the

set of all plain sets, so it is in itself, so it is fancy.

• If R is fancy it is not in the set of all plain sets, so it is not in itself, so it is plain.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …

What’s infinity plus 1?

0

1 2 3 4 5 6 7 8 9 10 11 …

1 2 3 4 5 6 7 8 9 10 110

So = +1

1,3,5,7,…

size

2,4,6,8,…

size ?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 …

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 …

1,3,5,7,…

size

2,4,6,8,…

size

So 2 x =

How many fractions are there?

22/7

…

….

….

….

… … … … … … …

So 2=

Set Cardinality

The whole numbersThe even numbers

All the fractionsAll the fractions between 0 and

1All the decimals between 0 and

1?

0 1

1/21/2 = 0.7…

0.101001001000100001…

If decimals fitted in the integer hotel:1 0.17396…2 0.09520…3 0.90024…4 0.31415…5 0.82274…… …

1 0.17396…2 0.09520…3 0.90077…4 0.31412…5 0.82274…… …

What about 0.20125…

Set Cardinality

The whole numbersThe even numbers

All the fractionsAll the fractions between 0 and

1All the decimals between 0 and

1c

(bigger than

Power sets

The power set is all subsets of a set

Power set of is

Power set of a set of cardinality n has cardinality 2n

A set can’t check its power set into its own hotel. If it could, see that some of the room numbers won’t be members of the set in their room. Consider the set of these room numbers, and find which room it is in…

3

0

1043

0, number of integers

c = 20, number of decimals

2c

3

0

1043

0, number of integers

1

2

ZF

c = 20, number of decimals?

“Inaccessibly infinite sets”

Is there a set whose cardinality is between the integers and the decimals?

The Answers

• + 1 = , kind of, but definitely 0 = 0+1

• 0 = 0+1 , 2 0 > 0

• Achilles catches the tortoise because some infinite sums converge to limits

• Some don’t, and 19C mathematics told us how to be sure which

• Infinite sets start getting really weird.

Les Fables d'Esope Phrygien, mises en Ryme Francoise. Auec la vie dudit Esope extraite de plusieurs autheurs par M. Antoine du Moulin Masconnois. A Lyon, Par Iean de Tournes, & Guillaume Gazeau. 1547. Fable 94. Du Lieure & de la Tortue. Flickr laura k gibbs

∑𝑖=1

𝑖=1012𝑖

=1− 1210

∑𝑖=1

𝑖=∝12𝑖

=1

• Hotel Infinity (Barrow Hilbert student story)– Infinite hotels ok

• Infinity as a process• Pascals wager• Infinity as a convenience; 1/0• Cantor’s diagonal argument• Hence Goedel and Turing…

– The Turing machine; uncomputables• Infinite bounded spaces; night sky paradox• Infinitely small the Arnold argument?• Escher for hyperbolic spaces

Symbol

• John Wallis 1655 from for M• Blake ‘hold infinity in the palm of your hand

and eternity in grain of sand’

1