Difference Between Zero and Nothing

Date: 12/12/96 at 10:49:18
From: Alistair Cockburn
Subject: The difference between 0 and nothing

My four-year old asked, "What is the difference between 0 and 
nothing?" and my tongue got all tied up.  How would you answer?

Thanks - Alistair Cockburn

Date: 12/12/96 at 23:05:40
From: Doctor Pete
Subject: Re: The difference between 0 and nothing

Well, first, I'm kind of surprised that a four-year old would even 
assume that there is indeed a difference between "zero" and "nothing," 
for it is not even clear to many older people (as you saw yourself) 
that these things are actually different concepts.  Most 
mathematicians consider 0 to be a number, and "nothing" to be the 
empty set; they are related in that the empty set has zero elements in 
it; that is, the *cardinality* of the empty set is zero.

To explain a bit more in detail, I will give a bit of set theory at 
this point.

We can think of "sets" as collections of objects.  For instance, we 
can have a set like:

     S = {dog, cat, horse, car}

I've used the braces "{ }" to group the objects together.  Each object 
(dog, for instance), is called an *element*.  Such a collection 
consists of "subcollections," or *subsets*.  That is, there is a 
subset of the above set which consists of those elements which are 
animals.  Mathematicians say:

     A = { x in S : x is an animal } = {dog, cat, horse}

We read this as, "A is the set of all x in S such that x is an 
animal."  So we say that A is *contained* in S.  Similarly, we can 
define another subset of S as:

     N = { x in S : x is a machine } = {car}

Or we could have said:

     N = { x in S : x is not an animal }
       = { x in S : x not in A }
       = S \ A

Here the backslash "\" is another notation mathematicians use, which 
is kind of like subtraction.  What happens is we let N consist of 
elements in S which are not in A.  Naturally, one might ask, what is

     E = { x in S : x is neither an animal nor a machine } ?

Or, if we really want to be crazy, what is

     E = S \ S ?

Well, it doesn't have any elements.  Such a set is called the empty 
set, which is written as "{}" or a zero with a slash through it.  Why 
this is not the same as 0 will become clear if we consider sets of 
numbers, rather than sets of objects.  For example, let

     S = {0, 1, 2, 3, 4}

What is the *cardinality* of S?  That is, how many elements does S 
have?  Clearly, it has 5.  Mathematicians write this as |S| = 5.  
Now, consider the subset {0} of S.  It contains a single element, 0.  
But it is not the empty set, for the empty set has *no* elements.  
Is the empty set a subset of S?  Sure!  To see why, ask yourself, 
"Is S a subset of itself?"  Yes, because S contains itself, or every 
element of S is also an element of S (of course).  Then S \ S must 
also be a subset of S.  But this is, of course, the empty set.  So 
both {} and {0} are valid subsets of S, but they are not the same.

To see an example of the difference between 0 and {}, we ask, "what is 
the value of x such that

     5 + x = 3 + 2 ?

Clearly, x = 0 is the answer.  Now, what about "what is the value of x 
such that:

     x + 5 = 1,  and  x + 1 = 1 

Obviously, there is no answer; that is, x = {}.

Now, hopefully, things are a bit more clear.  The idea of "nothing" 
stems from this notion of a collection.  Like eggs in a basket.  If 
you had no eggs (nothing in the basket), then this is analogous to the 
empty set.  The *number* of eggs in the basket is zero.  So we can 
think of "nothing" as a term describing the set itself, whereas "zero" 
is a term not describing a set, but an element.  The confusion between 
the two is a result of the fact that the number of elements in the 
empty set is 0.  There's a subtle difference in that, one that perhaps 
a 4-year old might have a problem understanding.  But it's definitely 
worth trying to explain.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/