Download - 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