Beyond Counting
Infinity and the Theory of Sets
Nate Jones & Chelsea Landis
Infinity
 Basis of Method of Exhaustion
 Used to find areas of curved regions
 Underlying idea of a limit
 Foundational concept of Calculus
 Relatively new to mathematics
 “…I protest above all against the use of an infinite
quantity as a completed one, which in
mathematics is never allowed. The Infinite is only
a matter of speaking…” Carl Friedrich Gauss
 Looking at any number we know that we
can always add 1 to any we come up
with
 Georg Cantor considered the collection
of all counting numbers as a distinct
mathematical object
 Rational and Irrational

(Ideas around during Civil War)
• Dense in each other
• Led to idea that real numbers were evenly
divided by the rational and irrational
Georg Cantor (1845-1918)
 Saw that rationals and irrationals are distinct
entities or “sets”
 Tried to compare rationals and irrationals and
tried to match them in a 1-1 correspondence
 Found that infinite sets could be compared
like finite sets.
 Cantor’s concept of a set:

By a set we are to understand any collection into a
whole of definite and separate objects of our
intuition or our thought.
Cantor’s Results
 Not all infinite sets are the same size.
 The set of irrationals is larger than the set of rationals
 Set of counting numbers is the same size as the set
of rationals
 The set of all subsets of a set is larger than the set
itself
 The set of points within any interval of the number
line, no matter how short, is the same size as the set
of all points everywhere on the number line
 The set of all points in a plane, or in 3-dimensional, or
(n-dimensional space) for any natural number n is the
same size as the set of points on a single line
Which infinity is greater?
Counting Numbers or Fractions?
There are just as many
counting numbers as
there are fractions!
Leopold Kronecker
 Prominent professor at the University of Berlin
 Disagreed with Cantor’s ideas of infinity
 His idea was that a mathematical object does not
exist unless it is actually constructible in a finite
number of steps
 Looked at the set of all even numbers that can be
written as the sum of 2 odd primes


This was never proven
This shows that if we can’t say what elements belong to the
set, how can we describe the set as a completed whole?
Bertrand Russell
(Paradoxes in Set Theory)
 A barber in a certain village claims he
shaves all those villagers and only
those villagers who do not shave
themselves. If his claim is true, does
the barber shave himself?
…
 If he’s in the set, he doesn’t shave himself,
but since he shaves all who don’t shave
themselves, that must mean he must shave
himself, so he ISN’T in the set.
 If he isn’t in the set, then he doesn’t shave
himself, but he only shaves those who don’t
shave themselves, so he must not shave
himself, so he IS in the set.
Set Theory (1874 - 1884)
 Provided a unifying approach to probability,
geometry, algebra, etc.
 Infinite sets were based on philosophical
assumptions. Cantor argued philosophically
his new ideas on mathematics.
 His works were looked at by mathematicians
and philosophers because at the same time
philosophers were looking for a way to
accommodate both science and religion.
 Mathematics can be done without first
resolving philosophical issues.
 Unlike Cantor, modern mathematicians
and philosophers, see the recognition of
the separation of math and philosophy
as a giant forward stride in the progress
of human thought.
 Neo-Thomism
 School of philosophical thought that viewed
religion and science as compatible
 Came about from Pope Leo XIII, in 1879, from his
writing of Aeterni Patris
 Held that science didn’t need to lead to atheism
and materialism.
 Cantor (Catholic) claimed infinite sets dealt
with reality, but they should not be mistaken
for the infinite God
Metaphysics
 The study of being and reality
 Cantor argued that infinite collections of
numbers had a real (not necessarily
material) existence.
 Neo-Thomistic philosophers in Germany
argued that because the Mind of God is all
knowing, God knows all natural numbers,
all rationals, all infinite decimals, etc.
Most Important effect of set theory in Philosophy
 Cantor’s investigations led to
clarifications of logical forms, methods
of proof, and errors of syntax. These
were used to refine arguments in
philosophy.
Georg Cantor’s Ideas
Even though from one point of view the entire
list of numbers we count with {1,2,3,4,5,.......}
is twice as large as the list of even numbers {2,4,6,8,10,.......},
the two lists can be matched-up in a one-to-one fashion.
This shows the two lists are the same size,
infinite.
Cantor’s Idea’s cont.
 Cantor was able to demonstrate that
there are different sizes of infinity.
 The
infinity of decimal numbers that are
bigger than zero but smaller than one is
greater than the infinity of counting
numbers.
• Cantor Diagonalization Proof
There are the same number of points on a short
semicircle arc as there are on the entire unbounded line.
Conclusion
 Cantor’s work has affected mathematics
in a positive way. His basic set theory
has provided a simple, unifying
approach to many different areas of
mathematics.
Timeline
 1774-1784 - Cantor’s work on set theory
 1879 - Pope Leo XIII wrote Aeterni
Patris
 1884 – Kronecker’s ideas came about
 Around 1919 - Paradoxes in Set Theory
 Early 20th century - investigation of
metaphysics and contradictions to set
theory
References
 Berlinghoff, William P, Gouvea, Fernando Q. Math
through the Ages A gentle History for Teachers and
Others. 1st edition. Farmington, Maine. Oxton House
Publishers, 2002.
 Counting to Infinity.
<http://scidiv.bcc.ctc.edu/Math/infinity.html> 11/27/06.
 Platonic Realms Minitexts. You can’t get there from
here.
<http://www.mathacademy.com/pr/minitext/infinity/ >
11/27/06.