Set

Definition of Natural Numbers



Subjects to be Learned

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The set of natural numbers can be defined using set theory. For this definition only set theory and logic are required.

Definition: the set of natural numbers N is the set that satisfies the following:

  1. Basis Clause: N
  2. Inductive Clause: If n N ,   then n { n } N.
  3. Extremal Clause: If S N and S satisfies 1 and 2 ,   then S = N
According to this definition, the set of natural numbers has as its elements the sets , { } , { , { } } ,
{ , { } , { , { } } } , ...  .
These are denoted as 0, 1, 2, 3, ...  .

Note that the definition of N given as an example of representation of set and recursive definition is not acceptable, because it uses the addition of natural numbers which requires the concept of natural number. Also it uses 0 without defining it.
The definition given here, on the other hand, relies only on the assumptions on set and logic.
Intuitively we can view these numbers as follows: first the empty set is a set that has nothing in it. Therefore it agrees with our intuition of number 0. We now have one concept which is the concept of zero. Thus construct a set that has 0 in it. It contains one object 0. That is the number 1. Now we have two distinct objects: 0 and 1. So construct a set that has these two objects. that is the number 2. Continuing in this manner, we can see the correspondence between the natural numbers and these sets.
This way we can characterize numbers without relying on anything except the concept of set and logic.

To appreciate this way of defining natural numbers, try to explain what the "oneness", "twoness" ... are. We seem to use "one", "two" ... with some object such as "one house", "two cars" etc. So if we want to explain (define) what "oneness" is, we can not seem to avoid using some object, say "house". Then we need to explain what that object (a house in this case) is. Otherwise you can't explain what "one house" means. You would have to answer questions like "Is a duplex a house or two houses ?", "How many buildings does a house have ? " etc. for example. On the other hand, "emptiness" deos not need any specific object for its definition. As everyone agrees, empty means there is no object there. Thus zero seems like a good place to start the definition of numbers. Once we accept the concept of "set" as given, we can thus combine that with "emptiness" and define "zero" as an empty set. We now have one object, an empty set. (The uniqueness of empty set can be easily proven). As shown above, number one can now be defined using the empty set, that is the set containing the empty set...





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