### In mathematics:

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.