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Infinity is the biggest thing you can imagine, then imagine a lot more. Infinity is a number, however it is very different from any normal number like 1, , 10, or 5 million, it is infinitely larger than any of these. ¡§But how big is infinity?¡¨ That is one question that will never be answered completely. For now let's start out with some simpler concepts and we will grow to that more difficult one later.

Let's think about some sets that we know are infinitely large; the set of positive integers, the set of real numbers, the set of even integers, the set of rationals, and so on. We know a lot of sets that are infinitely large. In fact if we think hard enough, we can come up with infinitely many, infinitely large sets. We know the set of integers that are multiples of one, the set of integers that are multiples of two, of three, and of any other integer. But this still doesn't answer our question ¡§how big is infinity?¡¨ Is any one of these infinities larger than any other?

One way to prove that two sets are the same size, or equinumerous, is to place them in a one-to-one match-up. Even without knowing how large two sets are we can prove they are the same size by pairing every element of one set up with a single element of the other set. If none are left over from either set, we know they are the same size. So how can we pair say all the positive integers with just the evens? Easy

1 „

„ 4

„ 6

„² „²

n „ n

Similarly we can pair all the integers with any multiple, x, of the integers, by saying n „ xn.

The sets we placed in a one-to-one match-up so far have all been countable, meaning they are the same size as the set of counting numbers (the natural numbers). Now lets try and put our same set of all the positive integers with the set of all rational numbers. This may be a bit more tricky considering that the rationals are considered a dense set, meaning in between any two are infinitely many more. If we can find a listing that contains all the rational numbers we can set it into a one-to-one match-up with the counting numbers. It turns out that this can be done, by listing the numbers in a table (or matrix) of fractions. Where each fraction is

Row number �n

Column number

Then list them starting at the position (1,1) and making a zigzag, tracing out (1,1), then (1,), then (,1), then (,1), then (,), then (1,), then (1,4), and so on until the entire table has been traced. Some duplicates will be found, however that is irrelevant because we have still found a well-ordered list of all the rationals. So it is true we can put the rationals into a one-to-one match-up with the counting numbers.

What about the reals? Well this is a bit harder than the rationals were, however Georg Cantor, found a way to prove that this cannot be done using his famous Diagonal Argument. It goes like this

1. Suppose there is a well-ordered list of the reals that includes all of them

. It can be set into a one-to-one match-up with the counting numbers

. Put all the reals into their decimal form

4. Now lets examine a specific number, call it x n is the counting number paired with a real. x is 0 + the nth decimal digit for each n. Let me clarify a little,

1 „ 1.45678¡K

„ .4715617¡K

„ .¡K

4 „ 0.00000000¡K

5 „ .141565¡K

„² „²

n „ another number

and so on

x is 0 + 0.00000 + 0.070000 + 0.00000 + 0.000000 + 0.00000 + ¡K

x = 0.70¡K

Basically if we line up the decimal points of all the reals in our list, x is all the decimal points along the diagonal with a zero whole part.

5. Lets look at another number y, which is each decimal point from x with one added to it

( becomes 0)

y is 0.48410¡K

6. y is not in our list because the first decimal place in y differs from the first number it cant be the first, and since the second decimal point differs from the second number it cant be the second, and so on for each number in our list. Therefore y is not in our list.

7. We have a contradiction, and thus the reals cannot be set into a one-to-one match-up with the counting numbers

We have now found a set that has a different size then the counting numbers, but both sets are infinitely large?!?!

What we have just determined is that infinity can have different sizes or levels. All of the sets I have mentioned thus far are of infinite size. The size of a set is called its cardinality. The sets {1}, {4}, {�à}, {e}, and {„w} all have cardinality one, the sets {1,0}, {e,7}, and {-.45, cos0} all have cardinality two. The counting set is given cardinality �ç (omega). The real numbers are given cardinality c (standing for continuum). The reason for this comes from the Continuum Hypothesis, which says that

If all subsets of a set have cardinality of the entire set or cardinality (call it x) smaller than the cardinality of the entire set, then the set in question's cardinality must be one higher than x.

The original intent of the continuum hypothesis was to say this

If all subsets of the reals have either a cardinality the same as the reals or positive integers then the cardinality of the reals is exactly one higher than that of the positive integers.

And so the term continuum was given to the real numbers and c was picked as its cardinality.

If the continuum hypothesis were true then c would be the next cardinality after �ç.

The cardinalities that we know of so far can be put in a set and ordered from least to greatest like this {1,,¡K�ç,?} We know �ç is the next one after all the counting numbers. Other than c is there anything else? If we look at Cantor's Theorem we can answer this question.

Cantor's Theorem

If X is any set, then there exists at least one set, the power set of X, called P(X), which has a higher cardinality than X. (The power set of X is formed by taking the set of all subsets of X)

Proof

1. Assume X is the largest set possible

. X = {a, b, c, d, e¡K}

. The power set of X is given by

P(X) = { {a}, {a, b}, {b, c, e}, {a, c}, {e} ¡K}

4. P(X) can't be smaller than X because it contains all the singletons that are in X, {a}, {b}, {c} etc.

5. P(X) can't be bigger than X because we assumed X was the largest set possible.

6. According to the Cantor-Schroeder-Bernstein Theorem

If cardinality of A is less than or equal to cardinality of B and the cardinality of B is less than or equal to the cardinality of A, then the two sets cardinalities are equal and a one-to-one match-up exists.

|A| „T |B| and |B| „T |A| then |A| = |B|.

The cardinality of P(X) equals the cardinality of X.

7. So we can create a one-to-one match-up

X P(X)

a „ a

b „ a, b

c „ b, c ,e

d „ a, c

e „ e

8. We can see that some X's appear in their match in P(X) on the right and some don't.

. Let's create a new set F that consists of all elements of X, which are not matched to subsets that contain them.

10. What element of X matches to F? If F is only the elements that don't match to things with themselves, no element of X can be matched to F, but F is a subset of X, which means that it must be in the power set of X. And so P(X) is really larger than X.

This tells us that there is no largest set. The Hebrew letter aleph, „q, has been adopted as a cardinal number. „q0 has been set to equal �ç. The reason for the „q's is because there are infinite cardinals that represent infinite sizes and by using subscripts it is possible to show this. Cantor wanted to call c (the continuum cardinal), „q1 (the next cardinality after „q0). This was the basis of his continuum hypothesis, however he was never able to prove his hypothesis.

To close out this paper I thought I might add a few fun things I came across in my research dealing with infinity and other topics I have discussed. This is the idea of paradoxes. One set that created problems was the set of everything thinkable. Cantor assumed there were two kinds of infinities in order to deal with the contradictions caused by some paradoxes. These two infinities were, as he called them, the consistent and the inconsistent infinities. Another paradox that I thought of, deals with the set of all sets (call it S). But S is a set and so by the definition of S, one of the elements of S must be itself. How can S be a set in which one of the elements of it, is itself? Another famous paradox I came across was Zeno's paradox. The problem was brought up in the form of a story where a tortoise challenged Achilles to a foot race. The tortoise claimed he could prove that if he was given any sized head start, say 10 feet he would win the race regardless of their speeds. His argument went like this. It will take you some time to catch up the 10 meters you are behind me, during that time I will have moved some distance, regardless of how small it is, forward. Now I still have a lead in the race. It will take a much smaller time for you to catch up the new distance, but it will take some time, in which I will advance further. Then I will still have a lead, and so by extending this I will always have some lead over you. Another way of examining the same question is how do we move form point A to point B? First we must go half way to B, and then from there, halfway again to B, and halfway again, and so on. We have an infinite amount of tasks to complete before we reach B. Which leads us to wonder whether motion is possible at all. Obviously we know motion is possible, so we look closer at the problem. We see how we are eventually adding infinitely small amounts of time. Say 1sec + 1/sec + 1/4sec +1/8sec +¡K 1/nsec = sec. This concept of adding infinitely many tasks is called a supertask. Some supertasks are possible, like this one, and others are not possible at all.

Infinity is a magical concept in mathematics and has always intrigued mathematicians in numerous ways. It has been a hotly debated topic in the mathematical world and has always remained a crucial part of many controversies in mathematics. This wonderful number will always be there, but will never be fully understood.

In the 10's Kurt Godel showed that the continuum hypothesis could never be disproved, based on the axioms of set theory. Later on in the 160's Paul Cohen showed that it cannot be proved, either. The continuum hypothesis is similar to the Axiom of Choice in this sense. The Axiom of Choice states that given two sets, one set has cardinality less than or equal to the other one. This also can never be disproved nor can it be proved. Therefore mathematicians are left to either accept it or reject it when conducting their work. By assuming the axiom of choice exists, it becomes possible to show that the continuum hypothesis can never be disproved. Barrow, John D. Pi in the Sky, 1. pp. 6-

http//www.ii.com/math/ch/#overview

http//www.mathacademy.com/platonic_realms/encyclop/articles/cantor_theorem.html

http//www.mathacademy.com/platonic_realms/encyclop/articles/zeno_tort.html

http//www.mathacademy.com/platonic_realms/minitext/infinity.html

http//www.math.lsa.umich.edu/~mathsch/courses/Infinity/Cardinality/Lesson1.shtml

http//www.math.lsa.umich.edu/~mathsch/courses/Infinity/Cardinality/Lesson.shtml

http//www.math.lsa.umich.edu/~mathsch/courses/Infinity/Cardinality/Lesson.shtml

http//www.math.lsa.umich.edu/~mathsch/courses/Infinity/Cardinality/Lesson5.shtml

http//www.thoralf.uwaterloo.ca/htdocs/scav/cantor/cantor.html

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