How Can Whole Numbers Be Defined Without Addition?

[MendelCyprys]

I'm not a mathematician of any sort so excuse me if my question is stupid.
I just realized that I could not define the set of whole numbers without referring back to them or to the operation of addition, which then itself can't be defined.
How would you define whole numbers?

 

[jedishrfu]

This is not a stupid question, its actually very deep and mathematicians pondered it for quite some time.

Wikipedia has an article on natural numbers that may help. About half way into the article is the formal (mathematical) definition.

http://en.wikipedia.org/wiki/Natural_number

 

[TylerH]

There's an easy way to recursively define them in terms of the sets of nested sets with only null sets or sets containing null sets.

Note that, if you have the first number and a way to get the next number given an arbitrary number, then you can construct the whole numbers. The function that takes a number and maps to its successor is called the successor function, S.

Let 0 = {}.
And let S(n)={n, {n}}.

So 1=S(0)={0, {0}} = {{},{{}}}.
2=S(1)={{{},{{}}},{{{},{{}}}}}.

It get's complex quick, so I won't do 3. You get the idea.

 

[jedishrfu]

There's an easy way to recursively define them in terms of the sets of nested sets with only null sets or sets containing null sets.

Note that, if you have the first number and a way to get the next number given an arbitrary number, then you can construct the whole numbers. The function that takes a number and maps to its successor is called the successor function, S.

Let 0 = {}.
And let S(n)={n, {n}}.

So 1=S(0)={0, {0}} = {{},{{}}}.
2=S(1)={{{},{{}}},{{{},{{}}}}}.

It get's complex quick, so I won't do 3. You get the idea.

Yes, more info on this approach from wikipedia:

http://en.wikipedia.org/wiki/Peano_axioms

Look for the set-theoretic models. The idea is to construct a collection of sets based on the empty and sets containing the empty set. The sets are constructed using a successor operation and have a correspondence with the whole numbers.