Download N Reals in (0,1)

Cantor and Countability:
A look at measuring infinities
How BIG is big?
10100 = a Googol
10googol = a Googolplex
109999999 = one Tremilliomilliotrecentrecentre
Mathematics in the 1800s:
Focus on Fundamentals
• Many of the fundamentals of mathematics were
reexamined in the 19th century.
▫ Major examples:
 Euclid’s parallel postulate
 The concept of a limit.
• Weierstrass developed a specific definition of the
limit.
• Much of calculus relied on different types of
numbers, and describing the nature of those
numbers.
• This is how Georg Cantor’s Set Theory was born.
Georg Cantor
An Original Mathematician
•1845 -1918
•Born in Russia
•Family moved to Germany while he was a
child, and this is where he spent most of his life
•Likely suffered from bipolar disorder –
hospitalized on several occasions
•Founder of “Set Theory”
•First to seriously consider infinities as
completed values
•Controversial figure in the mathematics
community of the time
•Believed understanding transfinite numbers
was a direct gift to him from God
Set Theory Basics
• Sets are fundamental groups of objects that
underlie mathematical thought.
• Two sets of objects are the same if they contain
exactly the same objects.
• Objects in sets are not repeated, and the order in
which they are present is not important.
• Each set contains a number of objects. This
number is called the set’s “cardinality”.
• The cardinality of a set S is written S .
Set Theory Basics:
Comparisons of Cardinalities
• Two sets that are not equal may still have equal
cardinalities.
• A <= B iff all of the elements of A can each be
mapped to one and only one in B.
• A >= B iff all of the elements of B can each be
mapped to one and only one in A.
• A = B iff both of the above are true.
Set Theory Basics: Countability
“Countably infinite” - the set can be arranged
ordinally.
In other words, the set has a 1-1 relationship with
the Natural numbers.
If two sets are countably infinite, we consider
them to be equal in size,
which is denoted as 0
(“aleph-naught”).
Set Theory Basics:
Examples of Countable Sets
1
2
3
4
5
6
4
6
8
10
12
…
N
…
5
3
1
2
4
Z
…
-2
-1
0
1
2
N
…
Even
2
…
…
Proof: Countability of Rationals
Set-up:
Can the set of all rational numbers Q can be
arranged in an order, thus having the same
number of elements (0) as N?
One may think there are more rationals than
positive integers, but using a very simple system,
we will prove the opposite.
We have to find some rule that sets up a 1-1
correspondence between N and Q.
Proof: Countability of Rationals
To start, Cantor made this clever chart:
0
In the chart at left,
1
1
2
2
3
3
4
...
1/ 2 1/ 2 2 / 2 2 / 2 3 / 2 3 / 2 4 / 2
...
any number in the ith
1/ 3 1/ 3 2 / 3 2 / 3 3 / 3 3 / 3 4 / 3
...
row has i as its
1/ 4 1/ 4 2 / 4 2 / 4 3 / 4 3 / 4 4 / 4
...
1/ 5 1/ 5 2 / 5 2 / 5 3 / 5 3 / 5 4 / 5
...
denominator and the
.
.
.
.
.
.
.
...
.
.
.
.
.
.
.
...
numerators are the
.
.
.
.
.
.
.
...
same through each
column, alternating to cover both negative and
positive rational numbers. 0 sits above the rest.
So, every rational number is on this array.
Proof: Countability of Rationals
Now, we trace a diagonal line through the chart,
0
skipping numbers
1
1
2
2
3
3
4
...
we’ve already found.
1/ 2 1/ 2 2 / 2 2 / 2 3 / 2 3 / 2 4 / 2
...
1/ 3
1/ 4
1/ 5
.
.
.
1/ 3
1/ 4
1/ 5
.
.
.
2/3
2/ 4
2/5
.
.
.
2 / 3
2 / 4
2 / 5
.
.
.
3/ 3
3/ 4
3/ 5
.
.
.
3 / 3
3 / 4
3 / 5
.
.
.
4/3
4/ 4
4/5
.
.
.
...
...
...
...
...
...
Proof: Countability of Rationals
Now, we trace a diagonal line through the chart,
0
skipping numbers
1
1
2
2
3
3
4
...
we’ve already found.
1/ 2 1/ 2 2 / 2 2 / 2 3 / 2 3 / 2 4 / 2
...
1/ 3
1/ 4
1/ 5
.
.
.
1/ 3
1/ 4
1/ 5
.
.
.
2/3
2/ 4
2/5
.
.
.
2 / 3
2 / 4
2 / 5
.
.
.
3/ 3
3/ 4
3/ 5
.
.
.
3 / 3
3 / 4
3 / 5
.
.
.
4/3
4/ 4
4/5
.
.
.
...
...
...
...
...
...
Proof: Countability of Rationals
Now, we trace a diagonal line through the chart,
0
skipping numbers
1
1
2
2
3
3
4
...
we’ve already found.
1/ 2 1/ 2 2 / 2 2 / 2 3 / 2 3 / 2 4 / 2
...
1/ 3
1/ 4
1/ 5
.
.
.
1/ 3
1/ 4
1/ 5
.
.
.
2/3
2/ 4
2/5
.
.
.
2 / 3
2 / 4
2 / 5
.
.
.
3/ 3
3/ 4
3/ 5
.
.
.
3 / 3
3 / 4
3 / 5
.
.
.
4/3
4/ 4
4/5
.
.
.
...
...
...
...
...
...
*Notice that 2/2 = 1
 it is skipped
Proof: Countability of Rationals
Now, we trace a diagonal line through the chart,
0
skipping numbers
1
1
2
2
3
3
4
...
we’ve already found.
1/ 2 1/ 2 2 / 2 2 / 2 3 / 2 3 / 2 4 / 2
...
1/ 3
1/ 4
1/ 5
.
.
.
1/ 3
1/ 4
1/ 5
.
.
.
2/3
2/ 4
2/5
.
.
.
2 / 3
2 / 4
2 / 5
.
.
.
3/ 3
3/ 4
3/ 5
.
.
.
3 / 3
3 / 4
3 / 5
.
.
.
4/3
4/ 4
4/5
.
.
.
...
...
...
...
...
...
And so on.
Now we can put these
numbers in the order
we found them in. 0 is first, 1 is 2nd, ½ is 3rd, and
so on.
Proof: Countability of Rationals
N
:1
2
3
4
5
6
7
Q
:0
1
1/2
-1
2
-1/2 1/3
…
…
With the order we’ve established, we have actually
chosen a 1-1 correspondence with the natural
numbers. Therefore, Q = N = 0.
Interlude
All even numbers, integers, and rational #’s are
countably infinite.
Is every infinite set of numbers countably infinite?
Interlude
NO!
In 1874, Cantor proved that this was not in fact the
case.
His original proof was a monster, but he revised it
in 1891, so here we present Cantor’s revised proof.
Proof: Uncountability of the Real
Numbers
Set-up:
Show that real numbers in the interval (0,1) are
uncountable, and uses this result to show that R
(the set of all real numbers) is uncountable.
Note:
So that each number has a unique representation,
we do not consider .x999999…, instead only
considering .(x+1)000000….
Proof: Uncountability of the Real
Numbers
Proof by contradiction:
Assume that the interval (0,1) can be put in a 1-1
correspondence with N.
N
Reals in (0,1)
1
<->
.1111111…
2
<->
.222222…
3
<->
.5
4
<->
.012345
…
<->
…
Proof: Uncountability of the Real
Numbers
Now, let’s think of a real number called b.
b is 0.b1b2b3… where the decimal values are chosen
as follows: Choose bn to differ from the nth place of
the number on the right side of our chart which
corresponds with n. However, the digit we choose
cannot be 0 or 9.
N
1
2
3
4
…
<->
<->
<->
<->
<->
Reals in (0,1) b = .2341… or .4827… , for example.
.1111111…
.222222…
.5
.012345
…
Proof: Uncountability of the Real
Numbers
Now, we know two things about b.
1. b is a real number. Less obviously, since we
couldn’t choose .00000 or .99999, b is not zero
or one. Thus, it is strictly within (0,1).
2. b cannot be one of the numbers on the righthand side of our chart, since it differs from each
in at least one place.
Proof: Uncountability of the Real
Numbers
#1 => b is on the right hand column of the chart.
#2 => b is not on the right hand side of the chart.
 This logical contradiction proves that our
assumption was wrong, and (0,1) can’t be put into
a 1-1 correspondence with N.
Proof: Uncountability of the Real
Numbers
Now we can use (0,1) like we were using N.
We’ll look for a 1-1 match with all real numbers.
Proof: Uncountability of the Real
Numbers
The easiest way to show that (0,1) has a 1-1
relationship with R is to find a function that only
exists on (0,1) and has asymptotes at each end.
(2 x  1)
For example, Cantor chose y 
2 .
(x  x )
Therefore, R is also uncountably
infinite!
y
(2 x  1)
(x  x2 )
1
Implications of This Discovery:
Implications of This Discovery:
N
Z
R
Q
Implications of This Discovery:
Not everybody was satisfied with his ideas. Some
people were hesitant to accept the idea of the
“completed infinity” on which Cantor’s ideas were
based.
However, by showing this fact about R, Cantor had
answered some of the pressing questions bothering
mathematicians of the day.
Implications of This Discovery:
For example:
Between any two numbers there are infinitely many
rational and infinitely many irrational numbers.
A function can be continuous except at rational points
But No function was continuous except at irrational points
There was some difference between the set of rational
numbers and the set of irrational numbers, but without
Cantor’s set theory, it wasn’t clear what was going on.
Implications of This Discovery:
At the time, very few transcendental numbers were known
to exist.
One may believe they were a relatively rare countable set.
Cantor was able to show transcendental numbers to be
uncountably infinite like he did with the irrational
numbers.
Sources
1. Journey Into Genius, Chapter 11
2. nndb.com/people (Picture of Georg Cantor)
3. http://math.boisestate.edu/~tconklin/MATH124/Main/Notes/
6%20Set%20Theory/PDFs/Cantor.pdf (Numbers from God)
4. http://en.wikipedia.org/wiki/Cantor's_first_uncountability_pr
oof (Cantor’s Original Proof)
5. http://www.math.wichita.edu/history/topics/numsys.html (random tidbit following this)
6. Dan Biebighauser (Constructible numbers image)
Random tidbit:
Next time you see someone holding up these hand
gestures:
Loudly Exclaim,
“Woooo! 4004!”
Random tidbit:
“Finger numerals were used by the ancient Greeks, Romans, Europeans of the
Middle Ages, and later the Asiatics. Still today you can see children learning to count
on our own finger numerical system. The old system is as follows:”
From Tobias Dantzig, Number: The Language of Science.
Macmillan Company, 1954, page 2.
As cited on http://www.math.wichita.edu/history/topics/num-sys.html