Grammatically, the word "nothing" is an indefinite pronoun, which means that it refers to something. One might argue that "nothing" is a concept, and since concepts are things, the concept of "nothing" itself is a thing. This logical fallacy is neatly demonstrated by an old joke that contains a fallacy of four terms: if nothing is worse than the Devil, and nothing is greater than God, then the Devil must be greater than God:

The Devil is greater than nothing.

Nothing is greater than God.

The Devil is greater than God.

Clauses can often be restated to avoid the appearance that "nothing" possesses an attribute. For example, the sentence "There is nothing in the basement" can be restated as "There is not one thing in the basement". "Nothing is missing" can be restated as "everything is present". Conversely, many fallacious conclusions follow from treating "nothing" as a noun. Modern logic made it possible to articulate these points coherently as intended, and many philosophers hold that the word "nothing" does not function as a noun, as there is no object that it refers to. There remain various opposing views, however — for example, that our understanding of the world rests essentially on noticing absences and lacks as well as presences, and that "nothing" and related words serve to indicate these.

small turkey, 0 is a number, and "nothing" is 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. We can think of "sets" as collections of objects. For instance, we can have a set like:

S = {dog, cat, horse, car}

We'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.