The definition of whole numbers

[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.

 

[blue_raver22]

I understand your concern about defining whole numbers without referring back to them or to the operation of addition. However, it is important to note that the concept of whole numbers is a fundamental aspect of mathematics and is often considered to be self-evident.

Whole numbers are a set of positive integers starting from zero and continuing indefinitely. They are used to represent quantities that are countable, such as the number of objects in a set or the position of an object in a sequence. Whole numbers can also be used in basic arithmetic operations such as addition, subtraction, multiplication, and division.

In terms of defining whole numbers, one approach is to describe them as the most basic building blocks of mathematics, upon which all other mathematical concepts and operations are built. In this sense, whole numbers can be seen as axioms or assumptions that are accepted without proof.

Another approach is to define whole numbers in terms of their properties. For example, whole numbers are always positive, they have no fractional or decimal parts, and they can be represented on a number line as equally spaced points.

Overall, while it may be challenging to define whole numbers without referring back to them or to the operation of addition, their importance and role in mathematics cannot be denied. I would encourage you to continue exploring and understanding the concept of whole numbers and their applications in various fields.