“TO INFINITY AND BEYOND”

A DEEPER LOOK AND UNDERSTANDING

INFINITY: DEF. AND SOME BACKGROUND

“To Infinity and Beyond” the phrase coined by Buzz Light-year in the Disney animated film Toy Story is somewhat redundant

Infinity as we know it is the “beyond”; it is as far, as large, as long, etc. as our brains can wrap around.

It can be defined as an unlimited extent of time, space, or quantity; eternity; boundlessness; immensity; or unlimited capacity, energy, excellence, or knowledge.

Natural numbers are said to be infinite: 1,2,3,…The number of points

on an unending line are infinite. Time and space are assumed infinite. With these observations it is easy to see why our friend Buzz’s phrase

is a little off base.

INFINITY: SOME HISTORY

In about the sixth century B.C., the first acknowledgement of infinity was made by the Greeks. They were the first to come up with philosophical sciences from practical ideas.

However, in a mathematical sense, they were not unable to “wrap their brains around” the concept so easily, so they sort of just dismissed the notion, and

avoided the concept . It was a long time later when infinity resurfaced in the sixteenth century with a

formula showing that pi can be calculated with other operations than geometry

ones. The formula was the first to use infinity to express a function. (See formula page #1)

Some History cont.

Another formula was exposed by the mathematician John Wallis in 1650. Wallis was born in 1616 in Ashford, Kent, England, and died in 1703. His formula included both pi and infinity as well. (see formula page #2)

A third formula uncovering an infinite series was detected by Gregory Leibniz in 1674. (see formula page # 3)

All three of these discoveries were the result of finding an approximation for pi. Since pi is in essence a never ending decimal, without the help of infinity, we would probably still be searching for a way to recognize and define the

expansion of pi. This was pretty much the standing point of infinity until later, with the genius of

George Cantor. He published the first of many papers to come about the concept of infinity.

INFINITY: BIOGRAPHY OF GEORGE CANTOR AND HIS CONTRIBUTION

George Cantor was a mathematician who was born in Russia, but lived in Germany most of his life. He was the son of a Danish merchant, George, and Maria, a Russian musician.

He attended German schools and received his Doctorate from the University of Berlin in 1867.

CANTOR BIO. Cont.

Cantor was a disciple of Immanuel Kent, and he was interested in developing a new logic of infinity, others than those used by astrologers, etc. before him.

He assumed there was something about rational numbers that makes them discontinuous, and too, something about real

numbers that made the set continuous.

Cantor cont.

Cantor worked to come up with a dichotomy between these two sets of numbers, and established a diagonal method (shown on paper) to demonstrate his theory.

He proved that the set of real numbers (R) is uncountable, and therefore larger than the set of all rational numbers (Q), with rational numbers being countable on the other hand.

Cantor saw infinity as a mixture of continuous, as well as discontinuous sets, while both being infinite still at the same time.

Cantor cont.

The last part of George Cantor’s life was not so productive.

He suffered from depression, which today would be considered as a case of bi-polar disorder.

He was hospitalized many times, and it was in one of the institutions that he passed away in 1918.

Although Cantor did not live to be very old, and his innovative mathematics faced much criticism, he was a stronghold force in the research and understanding of infinity, and so is remembered for that.

Infinity: Some of Cantor’s Work

Cantor used a one to one correspondence method, between two sets of numbers that have the same cardinality. If you use a finite set, and an infinite set, the one to one correspondence acts in the same way.

CORRESPONDENCE WITH A FINITE SET{ 1 2 3 4 5 } cardinality of five{ 4 4 5 6 8 } also cardinality of five(set) = (set)CORRESPONDENCE WITH AN INFINITE SETSet 1: {1 2 3 4 5 6...n…} (infinite number of counting numbers: cardinality = 0)Set 2: {0 1 2 3 4 5...n-1...} (infinite number of whole numbers: cardinality = 0)(set)=(set) Since corresponding numbers represented by “n” in the first set and “n-1” in

the second set, no matter what point is chosen in set one (n), (n-1) will represent the correct correspondence in set two, and both sets have cardinality zero.

More of Cantor’s Work

Cantor’s famous Diagonalization Proof shows that the cardinality of Reals can be proved to be larger than the cardinality of Natural numbers.

First, we assume that we have found a one to one correspondence between the reals and the natural numbers between zero and one.

Secondly, we put numbers in a list in decimal notation that we assume, again, to be complete.

Cantor’s Work cont.

1. .421341513654...2. .356745784568...3. .757456724253...4. .558783472955...5. .8789365276346. … Third, we assume our list is complete. Next, we choose numbers different than the first digit in the first

number, the second digit in the second number, the third digit in the third number, and so on, on down the diagonal.

For example we can choose our number to be .23452.… This “last” number will be different from the first in the first digit,

be different from the second digit in the second number, etc.

CANTOR’S WORK cont.

Therefore the number will differ from all numbers in the list, but is clearly a number between zero and one and is a real number.

Since our number is not in our list, our list is not complete as assumed. Since this is so, there cannot be any such one to one correspondence

between the reals and the naturals between zero and one. In this manner we can map the naturals onto the reals but not map the reals

onto the naturals. We can see through this that the infinity of decimal numbers that are bigger

than zero but less than one is greater than the infinity of counting numbers. No matter how large a set is, we can always consider a set that is larger…

infinity.

CONTRADICTION TO CANTOR

There have been many whom have questioned George Cantor’s reasoning for his infinity findings. Some have recognized mistakes that Cantor did not acknowledge, including the following argument.

CONTRADICTION TO CANTOR cont.

One of Cantor’s claims was that all non-finite countable sets have the same cardinality. (The contradiction starts out here)

If we have N = {0,1,2,3,4,…}There are sets I = {0,2,4,6,8...} (N x 2)II = {0,1,4,9,16...} (N squared)III = {1,3,5,7,9...} (N x 2 + 1) Claim I, II, and III have the same cardinality. Also goes on

to claim that I, II, III are proper subsets of the set N. This is where Cantor goes wrong…

Contradiction, cont.

The correct interpretation is that they are not subsets because they have the same cardinality as set N.

Cantor was unaware that given sets A,B: If “A sub B” (A is a proper subset of B) then

“Cardinality (A) < Cardinality (B)” Cardinality (A) = Cardinality (B) if and only if A 1-1 B

(there is only one element of set A for every element of set B).

Contemporary mathematicians of the time, including Cantor, were unable to imagine sets of greater cardinality than the set N.

CONCLUSION

The concept of infinity does not end with Cantor, obviously. His results were looked over and studied, and met with much criticism from other mathematicians, as well as his peers even.

As we all know, infinity still is prevalent today in physics, astronomy, cosmology, etc.

Modern mathematics makes much use of infinite sequences of numbers, and will probably in time come up with other ways to express this concept, considering the exponential ways of the subject.

Although some may consider, or even prefer, to not use infinity, or not try and understand it’s magnitude, we can always go beyond what infinity is intuitively.

But once redundant as in “To Infinity and Beyond”, ironically, the concept itself is a subject of the beyond.