Cantor%20and%20Countability:

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Cantor and Countability

• A look at measuring infinities
• How BIG is big?

10100 a Googol 10googol a
Googolplex 109999999 one Tremilliomilliotrecent
recentre
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Mathematics in the 1800s Focus on Fundamentals

• Many of the fundamentals of mathematics were
  reexamined in the 19th century.
• Major examples
• Euclids 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 Cantors Set Theory was born.

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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

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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 sets cardinality.
• The cardinality of a set S is written .

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Set Theory Basics Comparisons of Cardinalities

• Two sets that are not equal may still have equal
  cardinalities.
• lt iff all of the elements of A can
  each be mapped to one and only one in B.
• gt 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.

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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).

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Set Theory Basics Examples of Countable Sets

• N 1 2 3 4 5 6
• Even 2 4 6 8 10 12
• N 5 3 1 2 4
• Z -2 -1 0 1 2
• Note Countable sets can be combined to make
  another countable set.

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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.

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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.

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Proof Countability of Rationals

• Now, we trace a diagonal line through the chart,
  skipping numbers weve already found.

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Proof Countability of Rationals

• Now, we trace a diagonal line through the chart,
  skipping numbers weve already found.

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Proof Countability of Rationals

• Now, we trace a diagonal line through the chart,
  skipping numbers weve already found.
• Notice that 2/2 1 ? it is skipped

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Proof Countability of Rationals

• Now, we trace a diagonal line through the chart,
  skipping numbers weve 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.

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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 weve established, we have
  actually chosen a 1-1 correspondence with the
  natural numbers. Therefore, Q N ?0.

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Interlude

• All even numbers, integers, and rational s are
  countably infinite.
• Is every infinite set of numbers countably
  infinite?

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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 Cantors revised
  proof.

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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 .(x1)000000.
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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 lt-gt .1111111
• 2 lt-gt .222222
• 3 lt-gt .5
• 4 lt-gt .012345
• lt-gt

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Proof Uncountability of the Real Numbers

• Now, lets 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 Reals in (0,1) b .2341
  or .4827 , for example.
• 1 lt-gt .1111111
• 2 lt-gt .222222
• 3 lt-gt .5
• 4 lt-gt .012345
• lt-gt

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Proof Uncountability of the Real Numbers

• Now, we know two things about b.
• b is a real number. Less obviously, since we
  couldnt choose .00000 or .99999, b is not zero
  or one. Thus, it is strictly within (0,1).
• 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.

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Proof Uncountability of the Real Numbers

• 1 gt b is on the right hand column of the chart.
  2 gt b is not on the right hand side of the
  chart.
• ? This logical contradiction proves that our
  assumption was wrong, and (0,1) cant be put into
  a 1-1 correspondence with N.

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Proof Uncountability of the Real Numbers

• Now we can use (0,1) like we were using N.
• Well look for a 1-1 match with all real numbers.

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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.
• For example, Cantor chose
  .
• Therefore, R is also uncountably
• infinite!

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Implications of This Discovery
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Implications of This Discovery
R
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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 Cantors ideas were
  based.
• However, by showing this fact about R, Cantor had
  answered some of the pressing questions bothering
  mathematicians of the day.

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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 Cantors set theory, it
  wasnt clear what was going on.
•  

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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.

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Sources

• Journey Into Genius, Chapter 11
• nndb.com/people (Picture of Georg Cantor)
• http//math.boisestate.edu/tconklin/MATH124/Main/
  Notes/620Set20Theory/PDFs/Cantor.pdf (Numbers
  from God)
• http//en.wikipedia.org/wiki/Cantor's_first_uncoun
  tability_proof (Cantors Original Proof)
• http//www.math.wichita.edu/history/topics/num-sys
  .html (random tidbit following this)
• Dan Biebighauser (Constructible numbers image)

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Random tidbit

• Next time you see someone holding up these hand
  gestures
• Loudly Exclaim, Woooo! 4004!

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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/topic
s/num-sys.html