infinity — and beyond? · 2019-12-07 · infinity — and beyond? andrew brooke-taylor...
TRANSCRIPT
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!