Dale Oliver
Professor of Mathematics
Humboldt State University
PONDERING THE INFINITE
PLAN FOR TODAY

Thought Experiments
 Numbered
Ping Pong Balls
 Un-numbered Ping Pong Balls
 Zeno’s Paradox of Motion

Comparing Quantities without Counting
 Hotel

Cardinality
Transfinite Numbers
ANAXAGORAS OF CLAZOMENAE
“There is no smallest among the small and no
largest among the large; but always something
still smaller and something still larger.” (5th
century BC, Greece)
NUMBERED PING PONG BALLS
Scene: There is a very large barrel.
Immediately to the right of the
barrel on a slightly inclined track
we see an unending stream of
numbered Ping-Pong balls
ordered by number starting with
1, 2, 3, 4, and so on. There is
one Ping-Pong ball for each
natural number.
NUMBERED PING-PONG BALLS
This mental experiment will last for exactly 1
minute. Sixty seconds after we start, our
stopwatch alarm will beep, and we will stop.
Task 1: In the first 30 seconds, we pour the first
10 Ping-Pong balls (numbered 1 through 10)
into the large barrel. We then reach in the
barrel, find the Ping-Pong ball labeled “1”, and
remove it.
NUMBERED PING-PONG BALLS
Task 2: We now have 30 seconds remaining. In
the next 15 seconds, we dump the next 10
Ping-Pong balls (numbered 11 through 20), and
then reach in, find the ball numbered “2”, and
remove it.
NUMBERED PING-PONG BALLS
Task 3: We now have 15 seconds remaining. In
7.5 seconds, we quickly dump in the next 10
balls (numbered 21 through 30) and then find
and remove ball number 3.
Task 4: We now have 7.5 seconds remaining. In
3.75 seconds, we very quickly dump in the next
10 balls (#31 through 40) and then find and
remove ball #4.
NUMBERED PING-PONG BALLS
Remaining tasks: At each subsequent stage (n),
we dump 10 balls into the barrel (#10n-9
through #10n), find and remove ball #n, and do
this all in half the time of the prior stage.
Our experiment continues in a hurried and frantic
way. Soon, we must move faster than the
speed of light to accomplish our task.
Mercifully, the experiment ends after 60
seconds.
NUMBERED PING-PONG BALLS
After the exhausting experiment, we “look” into
the large Barrel. What do we see?
UNNUMBERED PING-PONG BALLS
Suppose we repeated the Ping-Pong Ball
experiment, this time with identical
unnumbered Ping-Pong balls. At each stage we
dump 10 balls into the barrel, and remove 1.
What do we “see” in the barrel after the
experiment?
ZENO OF ELEA (C. 490-435 BCE)
(c. 490-435 BCE)
A Pythagorean
Presented 40 Paradoxes to
challenge the assumption that
space and time could be
divided into arbitrarily small
pieces.
It was not until Cantor's
development (in the 1860's
and 1870's) of the theory of
infinite sets that the paradoxes
could be fully resolved.
ACHILLES AND THE TORTOISE
The Tortoise challenged Achilles to a 100 meter
race, claiming that he would win as long as
Achilles gave him a small head start. Achilles
laughed at this, for of course he was a mighty
warrior and swift of foot, whereas the Tortoise
was heavy and slow. “How big a head start do
you need?” he asked the Tortoise with a smile.
“Ten meters,” the latter replied.
ACHILLES AND THE TORTOISE
The paradox rests partly on the misconception
that an infinite number of ever-shorter lengths
(and similarly, time durations) must add up to
an infinite total.
But, an infinite series may have a finite sum.
WHEN WILL ACHILLES CATCH UP?
Suppose that Achilles runs 10 meters per
second, and the Tortoise runs 1 meter per
second. When does Achilles “catch” the
tortoise?
Achilles: Distance: 10+1+1/10+1/100+…
Time: 1+1/10+1/100+…
Tortoise: Distance: 1+1/10+1/100+…
Time: 1+1/10+1/100+…
CANTOR (1845-1918) AND CARDINALITY
Comparing without counting
Which has more?
@@@@@@@@@@
&&&&&&&&&&&&&
CANTOR AND COUNTING
Which has more?
1, 2, 3, 4, 5, 6, 7, 8,…
2, 3, 4, 5, 6, 7, 8, 9,…
CANTOR AND CARDINALITY
Which has more?
…-3, -2, -1, 0, 1, 2, 3, …
1, 2, 3, 4, …
CANTOR AND CARDINALITY
Two sets are equivalent if there exists a 1 to 1
correspondence between their elements.
A set is infinite if it is equivalent to a proper
subset of itself.
CANTOR
An infinite set is said to be countable if there is a
one-to-one correspondence between it and the
set of natural numbers (1, 2, 3, 4, …)
Hotel Cardinality (see handouts)
SOME THOUGHTS
“There are as many squares as there are numbers because they are just as
numerous as their roots.”
- Galileo (17th century)
“I can see it, but don’t believe it!”
- Georg Cantor (Late 19th – Early 20th century)
“God created the positive integers; all the rest is human creation.”
- Leopold Kronecker (Late 19th –Early 20th century)
CANTOR’S THEOREM
There are
more real
numbers
than
natural
numbers.
MORE QUESTIONS
1.
2.
3.
4.
Is there an infinity between the natural
numbers and the real numbers?
Is there an infinity greater than the infinity of
the real numbers?
Are there infinitely many different sizes of
infinity?
Is there a largest infinity?
VI HART INFINITY ELEPHANTS
http://vihart.com/doodling/infinityelephants.mp4