Dale Oliver Professor of Mathematics Humboldt State University PONDERING THE INFINITE

Transcription

1 Dale Oliver Professor of Mathematics Humboldt State University PONDERING THE INFINITE

2 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

3 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. (5 th century BC, Greece)

4 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.

5 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.

6 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.

7 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.

8 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.

9 NUMBERED PING-PONG BALLS After the exhausting experiment, we look into the large Barrel. What do we see?

10 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?

11 ZENO OF ELEA (C BCE) (c 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.

12 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.

13 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.

14 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+1/100+ Time: 1+1/10+1/100+ Tortoise: Distance: 1+1/10+1/100+ Time: 1+1/10+1/100+

15 CANTOR ( ) AND CARDINALITY Comparing without counting Which has more? &&&&&&&&&&&&&

16 CANTOR AND COUNTING Which has more? 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9,

17 CANTOR AND CARDINALITY Which has more? -3, -2, -1, 0, 1, 2, 3, 1, 2, 3, 4,

18 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.

19 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)

20 SOME THOUGHTS There are as many squares as there are numbers because they are just as numerous as their roots. - Galileo (17 th century) I can see it, but don t believe it! - Georg Cantor (Late 19 th Early 20 th century) God created the positive integers; all the rest is human creation. - Leopold Kronecker (Late 19 th Early 20 th century)

21 CANTOR S THEOREM There are more real numbers than natural numbers.

22 MORE QUESTIONS 1. Is there an infinity between the natural numbers and the real numbers? 2. Is there an infinity greater than the infinity of the real numbers? 3. Are there infinitely many different sizes of infinity? 4. Is there a largest infinity?

23 VI HART INFINITY ELEPHANTS