Help me understand infinity

Question

I asked a math professor once about infinity and his answer puzzled me. I asked if i had two sets of numbers:

A = all the whole numbers in infinity B = all the whole and half numbers (1, 1.5, 2, 2.5 etc...)

Is B twice as large as A? He chuckled and said no, and then explained something about ordinal and cardinal numbers, that left me confused and still not understanding why my logic that it would be twice as large (and not the same size as he argued) did not hold up. Any good way to explain it to a non-mathematician?

Answer 1

Suppose that there is extraterrestrial civilization which has it own mathematics: suppose that they have a funny notation for a natural numbers: they denote one by $1$, two by $1.5$, three by $2$ and so on... So using this notation they think about which of Your two sets?

Answer 2

The exercise is to realize that it's not actually clear what "twice as large" would mean in this context. It implies you have some sort of measure of 'size', but what measure could make sense here?

So the point is to really reflect on how to meaningfully quantify the size of these sets -- or barring that, to at least give meaning to the idea of one set being "larger than", "the same size as" or "smaller than" the other.

While you're pondering this, it would also be instructive to think about comparing the sets

1 3 5 7  9 ...
2 4 6 8 10 ...

as well as the sets

1 2 4 5 7 8 10 11 ...
2 3 5 6 8 9 11 12 ...

Answer 3

Given your sets $A$ and $B$ you can find a bijective mapping $A \leftrightarrow B$.

That bijective mapping is given by

$$ a = 2b-1 \leftrightarrow b = \frac{a+1}{2} $$

For every $a \in A$ there is a unique $b \in B$ and also for every $b \in B$ there is a unique $a \in A$.

So neither set can be larger then the other.

Answer 4

A slight variation on Hilbert's Hotel answers your question.

Imagine a hotel with infinitely many rooms, numbered 1, 2, 3, .... This gives one room for each "whole number" that you mention. We now want to add the "half" numbers that you mention. We just put number $n$ into room $2n+1$. This means that number 1 is in room 1, number 1.5 is in room 2, number 2 is in room 3, number 2.5 is in room 4, and so on.

Note that we now have your "whole and half numbers" in the same rooms that used to store just the whole numbers. That shows that the size of the two sets {1, 2, 3, ...} equals the size of the set {1, 1.5, 2, 2.5, 3, ...}. We have "counted" the set, in the same way that I can count the children in my Physical Science class by matching each student to one of my fingers, so the number of students is the same as the number of fingers.

This is the idea of cardinality that can be expressed by cardinal numbers as discovered by Georg Cantor. Is this clear?

Answer 5

We will not explain about one to one correspondence, and why in that sense the two sets have the same "size." That has been done very well by others.

But there are many ways to make formal the intuitive notion of size. We give one that is consistent with your intuition. Because of laziness, the explanation we give uses more symbols than necessary.

For any $x\gt 1$, let $f(x)$ be the number of members of the set $A$ which are $\le x$. So for example, $f(9.3)=9$, since there are $9$ numbers in $A$ that are $\le 9.3$.

Let $g(x)$ be the number of members of the set $B$ that are $\le x$. So $g(9.3)=17$.

Note that $g(9.3)$ is approximately equal to twice $f(9.3)$. For large $x$ the ratio $\dfrac{g(x)}{f(x)}$ will be very close to $2$. To use formal language, $$\lim_{x\to\infty}\frac{g(x)}{f(x)}=2.$$

This gives a precise sense in which $B$ is twice as large as $A$.

But to reiterate, in the sense of cardinality, the two sets have the same size.

Answer 6

Imagine you have two baskets: One contains the whole numbers and the other contains the whole and half numbers. You are going to form pairs of numbers, taking one from each basket: (1, 1), (2, 1.5), (3, 2), (4, 2.5),... etc. You just paired each number in the first basket with each number in the second basket, and so both baskets contain the same amount of objects.