TLC

					How big (really) is infinity?
A definite answer through transfinite arithmetic


This workshop will be an informal introduction to transfinite arithmetic and the
questions related to the quantification of infinities. Basically, it will try to provide a
satisfactory answer to all those nagging questions we’ve all had - likely before
the age of six - about the size of infinity. Queries like “What is infinity plus one?”
or “What is infinity plus infinity?” or even “What is infinity times infinity?” It turns
out Georg Cantor, the founder of set theory, had all the answers! Even so, he
commented on one of his seminal proofs that “although I see it, I do not believe
it.” Prepare to see things that you will hardly believe.. and - by the same token –
learn a quality lesson in parenting.



Motivational Questions

If all humans are inherently equipped with the ability to count and to repeat the
process of “adding one more” ad infinitum – starting with the number 1, add 1 to 1 to
get 2, then add 1 to 2 to get 3, and so forth – then why are we seemingly incapable
of…

   1) …providing a satisfactory answer (or justification) to our kids when they
      ask: “What is infinity plus one?” or “What is infinity plus infinity?” or even
      “What is infinity times infinity?”

   2) …explaining such an absurdity as Zeno’s paradox, which basically states
      that motion through space is impossible or, equivalently, that motion is an
      illusion?

   3) …deciding whether there are more real numbers between 0 and 1 than
      there are rational numbers (i.e. fractions) altogether?



I - Historical Perspective

Ancient Greeks                The Greek philosopher Zeno of Elea proposes his
(5th century B.C.)            famous paradoxes of motion. These are rooted in
                              deep questions about the nature of time and space
                              and, also, in some misconceptions about infinity:
                              - Achilles and the Tortoise (you can never catch up!)
                              - The dichotomy paradox (you can’t even start!)
                              - The arrow paradox (you can’t even move!)
Ancient Romans              Use the symbol ∞ to represent 1000, a “big” number
                            to them.

Galileo of Pisa             Galileo states his famous “paradox" and begins the
(Early 17th century)        first modern line of inquiry towards the infinite by
                            writing: “infinity should obey a different arithmetic than
                            finite numbers.”

John Wallis                 The English mathematician proposes that ∞ stand
(1665)                      for infinity.

Newton / Leibnitz           Both mathematicians independently develop the
(Late 17th century)         calculus, which systematically assumes the existence
                            of “infinitesimal” processes.
                            Analysis - one of the pillars of modern mathematics -
                            is born!

Georg Cantor                An ambitious Russian/German mathematician completely
(1870’s)                    upsets the mathematical order by founding the theory
                            of sets and by putting forth his (heretic) transfinite
                            arithmetic.
                            His tremendous accomplishment is twofold: he has
                            single-handedly invented the language of
                            contemporary mathematics - set theory – and, in
                            addition to that, he has grounded the concept of
                            infinity on a firm logical foundation.
                            He was rewarded for all this with a lifetime of
                            controversy, a stifled career and, ultimately, a sad
                            history of mental health.
                            Now, however, he justly holds a very special place in
                            the pantheon of mathematics.

Kurt Gödel                  The brilliant Austrian logician Kurt Gödel shows
(1930’s)                    that the continuum hypothesis cannot be disproved
                            from the axioms of set theory…

Paul Cohen                  …and then the American mathematician Paul Cohen
(1960’s)                    shows that the continuum hypothesis cannot be
                            proved!


As it stands today, we have still not settled the question of the continuum!
II - Mathematical Perspective – Set Theory and Transfinite Arithmetic

As humans we have a fundamental ability to perceive the items that surround us
as collections and to categorize them into abstract groups. This explains why we
also seem to have an inherent drive to collect things.

Definitions

A set is any collection of well-defined, well-distinguished objects. These are
called the elements, or members, of the set. For a given set S, the number of
elements of S is called the cardinality of S. (Think of the cardinality as the “size”
of the set.) If S has a finite number of elements, it is called a finite set.
Otherwise, the set is called an infinite set.


Examples

1 - A set of 12 crystal champagne glasses (card = 12)
2 - A complete set of golf clubs (card = 13):
       {PW, SW, Putter, 1W, 3W, 5W, 3, 4, 5, 6, 7, 8, 9}
3 - The set of prime numbers smaller than 100 (card = 25)
       {2, 3, 5, 7, … , 89, 97}
4 – The set of counting numbers
       N = {1, 2, 3, … }


Question: What about the cardinality of infinite sets, like N in example 4?

This is where Cantor comes in to provide all the answers…


Cantor’s Definition of an Infinite Set:
A collection is infinite, if some of its parts are as big as the whole.

Alternative Definition:
A set is infinite if it can be put in a one-to-one correspondence with a proper
subset of itself.


Cantor proposed that, since we can’t count the number of elements for any two
infinite sets, we should instead compare whether they have the same “size” (i.e.
are equinumerous) by seeking a one-to-one match-up between the elements of
the two sets.
Example:




Thus, the example above illustrates that the set of even numbers is
equinumerous to the set of counting numbers, contrary to what our intuition
would otherwise suggest (one should have twice as many elements as the other,
right? No, that’s wrong!)

The cardinality of the set of counting numbers (or even numbers) is called aleph-
null and it actually constitutes the smallest type of infinity.

Now, what can you say about the cardinalities of…

1)     …the set of integers?
2)     …the set of rationals?
3)     …the set of reals?

Are they all equal to aleph-null? How can you show if they have a different
cardinality?

Cantor answered this fundamental question and – in the process – posed one of
the most important queries in the development of mathematics:
the continuum hypothesis.


CANTOR’S DIAGONALIZATION PROOF
See hand-out.


CONTINUUM HYPOTHESIS

Theorem (Cantor)
If X is any set, then there exists at least one set, the power set of X, which is
cardinally larger than X.

Set of “cardinals” = {0, 1, 2, 3, … , aleph-null, aleph-one, aleph-two, … }

Let c denote the cardinality of the continuum (i.e. all real numbers on a segment
of the real line). Then, is it true that c = aleph-null ? The affirmative answer is
what is called the continuum hypothesis.

				
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posted:11/8/2011
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