The Concept of Infinity

1

• The Concept of Infinity
• In the History of
• Mathematics and its
• Implications

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The Pythagoreans

• Seemed to have some concept of infinity as they
  considered one the generator of all numbers. Any
  natural number no matter how big could be made
  bigger by adding one.
• Later showed an intolerance for the concept that
  there were numbers outside the natural numbers
  when it was discovered that the hypotenuse of an
  isosceles right triangle is not commensurate with
  its side. (Hippasus thrown from boat)
• Curriculum connection number systems
• - Gr. 11 U or M

3
Zenos Paradoxes

• One such paradox involving the concept of
  infinity is that of crossing a room. Suppose you
  are trying to get from one end of a room to the
  other by going half the distance to the other
  side each time. As in the paradox by Zeno, the
  distance remaining will keep shrinking but you
  will never reach the other side in any number of
  finite steps.
• Curriculum connection asymptotes/limits
  functions/calculus

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Method of Exhaustion

• Used by Eudoxus and Archimedes to determine the
  areas of 2D and 3D figures.
• A circle can be viewed as an infinite polygon.
• Curriculum connection meaning of p any level

5
Calculus

• It would be approximately 2000 years until the
  use potential infinity combined with an extension
  of these methods was used to develop the concept
  of a limit and calculus in general. Why did it
  take so long?

6
Causes for Lack of Progress

• Wars caused people to put their resources into
  surviving rather than advancing
• Narrow mindedness of the Catholic Church

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Church Doctrine

• Aristotelian thinking became intertwined with
  Church doctrine due to St. Thomas Aquinas
• Aristotle believed the natural numbers are
  potentially infinite because they have no
  greatest member. However, he would not allow that
  they are actually infinite, as he believed it
  impossible to imagine the entire collection of
  natural numbers as a completed thing.
• Aquinas believed that nowhere in creation was
  there an actual mathematical infinite, either as
  a magnitude (a property of a single thing) or as
  a multitude (a property of a plurality of
  things). But he accepted the potential infinite,
  in essentially Aristotelian terms and relying on
  essentially Aristotelian arguments.

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Galileos Paradox

• Galileo although famous for his work in other
  areas, turned to math and the infinite after he
  was forced under house arrest by the church
• Paradox When put into a one-to-one
  correspondence we can see that there are as many
  squares as natural numbers even though the number
  of squares is a proper subset of the natural
  numbers
• He came close but stopped just short of dealing
  with actual infinity
• Curriculum connection Inverses Gr. 11

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Bolzano

• A one-to-one correspondence can be made for
  closed intervals of different sizes. For example,
  using the function y2x we can map every x in the
  domain 0 to 1 uniquely to a y in the range 0 to
  2. This implies that there are many numbers on
  the closed interval 0 to 2 as there are in the
  closed interval 0 to 1 which is a proper subset.
  Similarly, it can be shown that there are as many
  numbers in any two closed intervals of different
  sizes.

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Cantor

• The most significant figure in the history of
  mathematical infinity. Cantor completely
  contradicted the Aristotelian doctrine
  proscribing actual completed infinities.
• The natural numbers are infinite but are
  considered countable. Cantor recognized the
  importance of one-to-one correspondence in
  showing whether any infinite set is countable.

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Cantors Diagonalization Proof of the
Denumerability of the Rational Numbers

• The set of rational numbers appears to be more
  dense but this method establishes a one-to-one
  correspondence.

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Cantors diagonalization argument

• Cantor proved by contradiction that the real
  numbers are non-denumerable.
• Assuming this list is complete leads to a
  contradiction as a new number can be created that
  is not on the list.

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Transfinite Cardinals

• We now have an infinite set of different
  cardinality than the other infinite sets we have
  examined and thus Cantor also showed the
  existence of different types of infinity. Cantor
  introduced the Jewish letter aleph, ?, to denote
  his different orders of infinity. Cantor believed
  there were a series of alephs ?0, ?1, ?2, ?3,
  to represent the transfinite cardinals. ?0 was
  the lowest infinite cardinal therefore it was the
  cardinality of the natural numbers. A few
  questions remained How many alephs are there?
  What was the cardinality of the continuum? Was it
  the next highest cardinal ?1, or some other
  cardinal?

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Transfinite Arithmetic

• ?m ?n ?n for all n m.
• ?m x ?n ?n for all n m.
• ?n lt 2 ?n for all n?N.
• The first two results show that addition and
  multiplication do not increase the cardinality of
  infinity. For example, adding the odd integers
  and even integers both of cardinality aleph null
  is equal to the set of all integers which still
  has cardinality aleph null. The interesting
  result above is that exponentiation increases the
  cardinality of a set. Put another way, the power
  set of a set, always has a higher cardinality
  than the set itself. This shows that there is no
  highest cardinal and therefore the set of
  infinite cardinals is itself infinite.

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The Continuum Hypothesis

• With this result for power sets, the question of
  the cardinality of the continuum could now be
  examined. Every number on the continuum has an
  infinite decimal expansion. Number systems using
  different bases are interchangeable therefore we
  can consider the binary expansion of any number
  on the continuum. At any position, there is
  either a 0 or a 1, and the number of positions is
  countably infinite, therefore the cardinality of
  the continuum, c, is 2?0.
• Cantor believed that the continuum was the next
  cardinality after that of the natural numbers, or
  2?0 ?1.

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Gods Messenger

• Mathematics, the role of infinity in particular,
  as seen in the light of Augustines
  writings(which were Neoplatonistic as opposed to
  Aristotelian), gained importance during this
  period as a way to understand God. Cantor
  believed in his different transfinite cardinals
  and thought the highest of these the Absolute,
  was God Himself. He also believed he needed no
  proof that these transfinite numbers existed
  because God told him so.
• Cantor had made the continuum hypothesis a matter
  of dogma. He no longer requred the proof that had
  eluded him for so many years. To him the
  assumption that 2?0 ?1 was not a statement
  that had to be proved. It was the word of God.

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Mental Illness

• At first Cantor believed the continuum hypothesis
  was true but the flip flopped back and forth
  between trying to prove it true then false. This
  was likely a major cause of his mental breakdown.
  Other potential causes were his relationship
  with his father, his realization of the forbidden
  nature of the knowledge he was seeking,
  opposition from Kronecker which cause him
  difficulties in the publishing and acceptance of
  his work.

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Zermelo

• Zermelo then went on to build on work and
  axiomatize set theory. To protect his continuum
  hypothesis from attack, Cantor had to prove every
  infinite cardinal was one of the alephs. To do
  this he needed to prove the well-ordering
  principle that every set can be well ordered. A
  set is well ordered if every one of its non-empty
  subsets has a smallest element. Cantor was not
  able to prove this but luckily Zermelo came to
  the rescue and did. In his proof, Zermelo used
  what he called the axiom of choice. Other
  mathematicians had issues with this axiom but it
  turned out in the end that the axiom of choice
  was equivalent to the well-ordering principle.

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New Paradoxes

• Hilberts Infinite Hotel In an infinite hotel
  which is full, one more guest can be
  accomodated by moving everyone down one room and
  an infinite number of guests can be accomodated
  by moving all the occupants to the even rooms
  thus leaving the odd rooms vacant.
• Russells paradox Russell considered the set of
  all sets that are not members of themselves.
  Russell called this set R. Then he asked Is R a
  member of itself? Here, Russell obtained a
  paradox. If the set R is a member of itself then
  it isnt. And if R is not a member of itself,
  then it is.

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Gödel and Cohen

• Like Cantor, when Kurt Gödel began to touch the
  forbidden concepts of the alephs and actual
  infinity, he too became mentally ill. He then
  began trying to design a mathematical proof of
  the existence of God. Gödel was also half way to
  proving that the continuum hypothesis was
  independent of the rest of mathematics but gave
  that up in the midst of his mental illness. Paul
  Cohen completed the other half of the proof Gödel
  started and thus that the continuum hypothesis
  was independent from all the axioms within the
  current Zermelo-Fraenkel axiomatic system. The
  continuum hypothesis can neither be proven true
  or false.

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Conclusion

• The concept of infinity has profound
  philosophical implications, even when being dealt
  with as a purely mathematical concept. Throughout
  history, it has been surrounded by controversy
  and has been immensely difficult for those who
  dared to confront it.