Counting
J.M.Basilla
To Infinity and Beyond
Math 1
General Education Mathematics
Lecture 9
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Julius Magalona Basilla
Institute of Mathematics
University of the Philippines-Diliman
[email protected]
2010 Math 1
Three unrelated questions
Counting
J.M.Basilla
Is zero a counting
number?
1. Is zero a counting number?
2. Is atheism a religion?
3. Is anarchy a form of government?
Most likely if you answer yes to one of these
questions, your answer to the others will be yes?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
0 is not a counting number.
Counting
J.M.Basilla
Is zero a counting
number?
1. Count (definition www.thefreedictionary.com)- to
name or list (the units of a group or collection) one by
one in order to determine a total.
2. Counting numbers - the numbers used in
reciting/counting .
3. zero is never recited when counting.
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
0 is a counting number.
Counting
J.M.Basilla
How many dolphins do you see inside this
room?
After counting the number of dophins in this
room(which is not existing), we knew that there are
zero dolphins in the room. Therefore, 0 is a counting
number.
If 0 is a counting number, then atheism is a religion
and anarchy is a form of government.
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Resolution
Counting
J.M.Basilla
The problem lies with the English language rather
than philosophical.
To date, even mathematicians recognize the fact that
the question whether 0 is a counting number or not is
undecidable.
Care is extremely practiced by every mathematician
in dealing with this question. Often times books and
papers state whether it takes 0 as a counting number
or not.
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
How large can large be?
Counting
J.M.Basilla
1. Approximately how many number of cells are there in
the human body?
more than 1014 or 100, 000, 000, 000, 000.
2. How many atoms are there in one mole? (chemistry)
6.022 ×
000,000
1023
or 602, 200, 000, 000, 000, 000,
3. How many stars are there in the universe?
about 5 × 1022 or 50,000, 000, 000, 000, 000, 000,
000
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Name for large quantities
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
1.
http://en.wikipedia.org/wiki/Names_of_large_numbers;
http://www.unc.edu/ rowlett/units/large.html
2. googol = 1 ×
10100 ,
3. googolplex = 1 ×
zeros.
i.e. 1 followed by 100 zeros
100
1010
or 1 followed by a googol
To infinity and
beyond
The Aleph ℵ
To infinity and beyond
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
To infinity and beyond
Counting
J.M.Basilla
Infinity, or ∞ in symbol is perhaps the most abused
mathematical term.
We think of infinity when we ponder on questions like
I
I
I
I
How many stars are there in the sky?
How many descendants will Abraham have?
How many counting numbers are there?
How many real numbers(decimal numbers) are
there?
We speak of gods having ”infinite” powers and
capabilities.
Today, we should ask ourselves: ”Do we really know
what infinity means?” and ”How much would be
infinite?”
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
How is this possible?
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
On a dark desert highway – not much scenery
Except this long hotel stretching far as I could see.
Neon sign in front read ”No Vacancy,”
But it was late and I was tired, so I went inside to plea.
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
The clerk said,”No problem. Here’s what can be done–
We’ll move those in a room to the next higher one.
That will free up the first room and that’s where you can
stay.
I tried understanding this as I heard him say:
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
CHORUS: Welcome to the HOTEL INFINITY –
Where every room is full (every room is full)
Yet there’s room for more.
Yeah, plenty of room at the HOTEL INFINITY –
Move them down the floor (move them down the floor)
To make room for more.
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
I’d just gotten settled, I’d finally unpacked
When I saw 8 more cars pull into the back.
I had to move to room 9; others moved up 8 rooms as
well.
Never more will I confuse a Hilton with a Hilbert Hotel!
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
My mind got more twisted when I saw a bus without end
With an infinite number of riders coming up to check in.
”Relax,” said the nightman. ”Here’s what we’ll do
Move to the double of your room number:
that frees the odd-numbered rooms.”
(Repeat Chorus)
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Hilbert Hotel
Counting
J.M.Basilla
Is zero a counting
number?
Last thing I remember at the end of my stay–
It was time to pay the bill but I had no means to pay.
The man in 19 smiled, ”Your bill is on me.
20 pays mine, and so.”
lyrics by Lawrence Mark Lesser.
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
Moral Lesson From Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
1. The are as many counting numbers as there are
whole numbers.
1
2
3
4
5
6
7
8
The concept of
Large Numbers
9
· · · To infinity and
beyond
The Aleph ℵ
0
1
2
3
4
5
6
7
8
···
Moral Lesson From Hilbert Hotel Problem
Counting
J.M.Basilla
Is zero a counting
number?
2. There are as many counting numbers, as there are
multiples of 2
1
2
3
4
5
6
7
8
9
The concept of
Large Numbers
· · · To infinity and
beyond
The Aleph ℵ
2
4
6
8
10
12
14
16
18
···
The real numbers between 0 and 1 and the
entire numbers
Which one has more? The numbers between 0 and
1, or the entire real numbers.
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
0.25 0.75
0
0.5 1
−∞
0
The Aleph ℵ
∞
There are as many numbers between 0 and 1 and as
there are numbers.
The cardinality of numbers between 0 and 1
Is there another sense of infinity?
Consider the following list of numbers between 0 and
1.
r1 = 0.1855845858 . . .
r2 = 0.274857485457 . . .
r3 = 0.8248425457478 . . .
r4 = 0.248545485724857857 . . .
r5 = 0.474854785475847587 . . .
.. ..
.=.
Can x = 0.28566 . . . (on the ith position we choose a number
different from the ith digit of ri be on the list? No!)
We can do this for any list and hence, there are more
real numbers between 0 and 1 than the counting
numbers.
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
To infinity and
beyond
The Aleph ℵ
The Aleph
Counting
J.M.Basilla
Is zero a counting
number?
The concept of
Large Numbers
0 : The cardinality of the empty set
1 : The cardinality of the set containing one more
element than the emptyset
ℵ0 : The cardinality of the set of counting numbers.
To infinity and
beyond
The Aleph ℵ
Properties of ℵ0
Counting
one more than an ℵ0 is still ℵ0 .
J.M.Basilla
For any natural number n, n + ℵ0 = ℵ0 .
Is zero a counting
number?
twice an ℵ0 is still ℵ0
The concept of
Large Numbers
For any natural number n, nℵ0 = ℵ0 .
To infinity and
beyond
ℵ0 × ℵ 0 =
ℵ20
= ℵ0 .
The Aleph ℵ
However, ℵ0 × ℵ0 × · · · × ℵ0 > ℵ0 .
{z
}
|
ℵ0 times
ℵ0 × ℵ0 × · · · × ℵ0 = ℵ1 .
{z
}
|
ℵ0 times
Continuum Hypothesis Is there an infinite number ℵ
such that ℵ0 < ℵ < ℵ1 undecidable