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We may define the successor of a set as follows.
- Let X be a set, and let me denote the union of two sets X and
Y as XUY (because special characters show up weird in
some browsers).
- The successor of X is s(X)=XU{X}.
- Remember, in set theory X and {X} are different.
To get an idea of what that definition of
successor says,
- if X={a,b,c},
- then s(X)={a,b,c,X}={a,b,c,{a,b,c}}.
- We can go farther and get the successor of the successor of X as
s(s(X))=s(X)U{s(X)}={a,b,c,X,s(X)}={a,b,c,X,{a,b,c,X}}.
While delightfully annoying to write down, this may seem kind of
crazy and pointless, but it actually isn't. The reason they call
this the successor is that you can represent the natural numbers through sets,
and then you can define what you would normally think of as the successor of a natural
number (i.e. 2 is the successor of 1, because it comes after 1). In this way you can
prove from set theory the Peano postulates for the natural numbers (which define
their properties). If you're really a geek, read on and I'll describe the
beginning of how this works. Let me represent the null set with the word null,
and, to clarify, the null set is the set with no members. Ok, from the axioms of
set theory, we have:
- null exists
- For any set X, {X} is not equal to X.
So basically, you can give null the new name zero. You can then
get the successor of null.
- s(null)=nullU{null}={null}
because, remember, null has nothing in it, so those pesky terms of
a, b, c and the like aren't there, but still
{null} isn't the same as null according to the axioms we have.
So, you can call the successor of null the successor of zero and give it the
new name "one". You know that null exists by definition, and if it
exists, you've just shown that the successor exists. You may continue this to get a
definition for any number you want. For two:
- s(s(null))=s({null})={null,{null}}
Again, I know this may seem kind of nuts, but in this way you can, with a lot of other
somewhat complex proofs, extend this idea of the successor to define the
operations on the natural numbers (like addition and multiplication). So,
the point is that from the axioms of set theory you can define and prove the existence of a
set which satisfies the Peano postulates, meaning you've defined the natural numbers.
Once you have those, with a few other axioms from set theory you can define the
real numbers. While very strange and abstract, it's impressive that you
can define and prove the existence of the real numbers just from the idea of sets.
At the suggestion of JerboaKolinowski, see also ordinal.
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