How Can a Set Be Empty?

Date: 09/29/2003 at 01:27:50
From: Hash
Subject: Why is empty set called a set ?

Why is the empty or null set called a set when it has no elements?  Is
there a mathematical proof that it's a set? 

Date: 09/29/2003 at 08:33:38
From: Doctor Peterson
Subject: Re: Why is empty set called a set ?

Hi, Hash.

We try to make our definitions so that they are as useful as possible. 
In this case, we would like all the operations we can do between sets 
to yield sets, just as we want addition and multiplication of two 
numbers to produce a number. Now, what happens when you take the 
intersection of a pair of disjoint sets (sets with no elements in 
common)? The result is an empty set, right? If we didn't call that a 
set, then in this (rather common) case, the result of the intersection 
operation would not be a set.

This is typical of the way math is done. We make some natural 
definition (for example, thinking of a set as any collection of 
objects), and then work with it; eventually we find that we have to 
refine our definitions, or clarify the extreme cases, in order to make 
our new branch of mathematics work neatly. We can't "prove" that the 
empty set is a set, since we are defining it as such; but we do have 
to demonstrate that it is a useful and consistent definition that 
produces interesting mathematics. It does!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/