Subject: geometry

name: Carlos

Iam: student

Can a infinite set be smaller than another infinite set? If so why?

Hi Carlos,

Infinity is something that no living person has ever seen, so naturally, there are different ways one might imagine it to be. The question is not so much which approach is correct, but which one is more useful. If you wish, you can simply say that infinity is not a number, but just a concept of being farther away than any place you can travel to, being larger than any number you can count to, coming after any time you can think of, and generally being beyond any limit. There is only one such infinity; since it is not a number, you can not do arithmetic with it.

However, there is a very different notion of infinity. In mathematics, it is often useful to work with the concept of "size." Size has the useful property of being comparable. When we compare finite sizes we often do so with numbers. If a theatre holds 300 people and 270 show up to see the movie, then we know that the number of seats is larger than the number of people since 300 is larger than 270. If you don't know the number of seats or the number of people you can still tell if there are more seats than people. Have everyone sit down. If there are seats left over then there are more seats than people.

This technique of matching the people with the seats is the essence of the method that mathematicians use to compare the sizes of sets, whether finite or infinite. It comes from Gerog Cantor, a mathematician who lived from 1845 to 1918.

Suppose that A and B are sets. A one-to-one correspondence between the sets is a matching of the elements of A and B so that each element of A is matched with one and only one element of B, and each element of B is matched with one and only one element of A. If you can construct a one-to-one correspondence between A and B then we say A and B are the same size. (More precisely A and B have the same cardinality.) If every attempt to construct a one-to-one correspondence between A and B, leaves B with elements that are not matched with elements of A, then we say that B is larger than A. (The cardinality of B is larger than the cardinality of A.) Hence, in the example above, the cardinality of the set of seats in the theatre is larger than the cardinality of the set of people, since every attempted arrangement of one person in each seat leaves some seats empty.

Now lets look at an infinite example. Suppose that A is the set of positive integers, 1, 2, 3, 4,... and B is the set of all real numbers between 0 and 1. I want to think of these numbers as decimal fractions, for example 0.75341123... I am going to show that no matter how you try to construct a one-to-one correspondence between A and B there are always members of B left over, and hence the cardinality of B is larger than the cardinality of A.

Suppose that I have A correspondence that I think works.

Integer<----->Real number
1<----->0.xxxxxxxxxx....
2<----->0.xxxxxxxxxx....
3<----->0.xxxxxxxxxx....
4<----->0.xxxxxxxxxx....
.<----->.
.<----->.
.<----->.
.<----->.

I am going to construct a real number 0.yyyyyyyyy.... in B which does not correspond to and integer in A, that is a real number in A that doesn't appear in the list on the right.

Look at the first real number 0.xxxxxxxxxx.... in the list.
• If its first digit after the decimal is 1 then I let the first digit after the decimal in 0.yyyyyyyyy.... be 3. If the first digit after the decimal is not 1 then I let the first digit after the decimal in 0.yyyyyyyyy.... be 1.

Now look at the second real number in the list.
• If its second digit after the decimal is 1 then I let the second digit after the decimal in 0.yyyyyyyyy.... be 3. If the second digit after the decimal is not 1 then I let the second digit after the decimal in 0.yyyyyyyyy.... be 1.
Next look at the third real number in the list.
• If its third digit after the decimal is 1 then I let the third digit after the decimal in 0.yyyyyyyyy.... be 3. If the third digit after the decimal is not 1 then I let the third digit after the decimal in 0.yyyyyyyyy.... be 1.
Continue in this fashion.

I claim that the real number I am constructing, 0.yyyyyyyyy...., does not correspond to any positive integer. The reason is that for any integer n the nth place after the decimal in 0.yyyyyyyyy...., is not the same as the nth place after the decimal in the real number that corresponds to n. Thus any attempted one-to-one correspondence between A and B always leaves some members of B left over. Hence "B is larger than A".

Cheers,
Chris and Penny
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