Cantor and Countability:

A look at measuring infinities

How BIG is big?

10100 = a Googol10googol = 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.• <= iff all of the elements of A can

each be mapped to one and only one in B.

• >= iff all of the elements of B can each be mapped to one and only one in A.

• = iff both of the above are true.

A B

A B

A B

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 …

Even 2 4 6 8 10 12 …

… 5 3 1 2 4 …

… -2 -1 0 1 2 …

Note: Countable sets can be combined to make another countable set.

Proof: Countability of Rationals

Set-up: Can the set of all rational numbers can be

arranged in an order, thus having the same number of elements (0) as ?

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 and .

Proof: Countability of Rationals

To start, Cantor made this clever chart:In the chart at left,any number in the ith

row has i as its 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.

..........

..........

..........

...5/45/35/35/25/25/15/1

...4/44/34/34/24/24/14/1

...3/43/33/33/23/23/13/1

...2/42/32/32/22/22/12/1

...4332211

0

Proof: Countability of Rationals

Now, we trace a diagonal line through the chart, skipping numbers we’ve already found.

..........

..........

..........

...5/45/35/35/25/25/15/1

...4/44/34/34/24/24/14/1

...3/43/33/33/23/23/13/1

...2/42/32/32/22/22/12/1

...4332211

0

Proof: Countability of Rationals

Now, we trace a diagonal line through the chart, skipping numbers we’ve already found.

..........

..........

..........

...5/45/35/35/25/25/15/1

...4/44/34/34/24/24/14/1

...3/43/33/33/23/23/13/1

...2/42/32/32/22/22/12/1

...4332211

0

Proof: Countability of Rationals

Now, we trace a diagonal line through the chart, skipping numbers we’ve already found.

*Notice that 2/2 = 1 it is

skipped

0

..........

..........

..........

...5/45/35/35/25/25/15/1

...4/44/34/34/24/24/14/1

...3/43/33/33/23/23/13/1

...2/42/32/32/22/22/12/1

...4332211

Proof: Countability of Rationals

Now, we trace a diagonal line through the chart, skipping numbers we’ve already found.

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.

0

..........

..........

..........

...5/45/35/35/25/25/15/1

...4/44/34/34/24/24/14/1

...3/43/33/33/23/23/13/1

...2/42/32/32/22/22/12/1

...4332211

Proof: Countability of Rationals

: 1 2 3 4 5 6 7 …

: 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, = = 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 NumbersSet-up: Show that real numbers in the interval (0,1) are uncountable, and uses this result to show that (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 NumbersProof by contradiction:

Assume that the interval (0,1) can be put in a 1-1 correspondence with .

Reals in (0,1)1 <-> .1111111…2 <-> .222222…3 <-> .54 <-> .012345… <-> …

Proof: Uncountability of the Real NumbersNow, 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. Reals in (0,1) b = .2341… or .4827… , for example.1 <-> .1111111…2 <-> .222222…3 <-> .54 <-> .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 right-hand 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 .

Proof: Uncountability of the Real Numbers

Now we can use (0,1) like we were using .

We’ll look for a 1-1 match with all real numbers.

Proof: Uncountability of the Real NumbersThe easiest way to show that (0,1) has a 1-1 relationship with is to find a function that only exists on (0,1) and has asymptotes at each end.

For example, Cantor chose .

Therefore, is also uncountablyinfinite!

)(

)12(2xx

xy

1

)(

)12(2xx

xy

Implications of This Discovery:

Implications of This Discovery:

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 , 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 pointsBut 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 112.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_proof (Cantor’s Original Proof)

5.http://www.math.wichita.edu/history/topics/num-sys.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:

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

“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:”