What is the rigorous definition of addition?

Question

I hope this is not a stupid question, but can we have a 'rigorous' definition of addition of two numbers? I am asking this because I haven't seen one. Almost all resources talk about addition being 'the sum of two numbers'. Understanding it becomes less intuitive as we go from $\mathbb{N}$ to $\mathbb{R}$. Suppose $a,b \in\mathbb{R}$. Then, instead of just using language to say that $a+b$ is the sum total of $a$ and $b$, can we define $a+b$ using mathematical notation, or is it just that $a+b$ is another member of $\mathbb{R}$? I hope I made my point clear.

Answer 1

The way how addition for reals is defined depends a bit on how one introduces the reals.

Constructive

Let us first look at the situation for natural numbers. If one is rigorous one defines addition recursively, where $s$ denotes the successor function.

For $n \in \mathbb{N}$:

• $n+0 = n$
• $n+s(m)= s(n+m)$

From this one can derive/prove the "usual" properties of addition, like commutativity etc. See the Wikipedia page on Peano axioms for more details.

Then one can proceed to construct the integers from the natural numbers, usually as equivalence classes of pairs of naturals, and one extends the definition of the addition to these equivalence classes of pairs (in this case just doing coordinate wise addition), showing that it is well-defined and again verifying it has the properties one wants.

Then one to rationals, basically the same procedure.

Then one has addition on the rationals. Now, if one constructs the reals as equivalence classes of Cauchy sequences then the addition of the reals is induced by (coordinatewise) addition of sequences of rationals, and thus addition of rationals (of course one always needs to check well-defined etc.)

If one introduces the reals via Dedekind cuts that is certain subsets of the rationals one can defines addition on those cuts, also reducing it to addition of rationals.

So one reduces addition of reals, to that of rationals, to that of integers, to that of naturals, which is defined recursively. The details depend on how one constructs the reals.

Axiomatic

One can say the reals are a/the totally ordered field that is Dedekind complete (i.e., every non-empty subset has a least upper bound), which is unique up to isomorphism.

If one does this one has addition by virtue of the structure being a field by definition. However at some point one likely will want to verify the existence of such a structure, which I think is usually done by the constructive approach above.

Answer 2

There are several possible equivalent ways to do this. If you want a rigorous definition of addition, you can

• start with one element, let us call it $0$ (one other option is to start with $1$)

• define the "successor" operator, noted $succ$

• define $\mathbb{N}$ thanks to $succ$

• define the addition in $\mathbb{N}$ thanks to $succ$.

• find out some properties of the addition thanks to those of $succ$

Then, if you want to go to $\mathbb{R}$ (defining $\mathbb{R}$ is always a long job), one way to go is :

• define the substraction

• define $\mathbb{Z}$

• define the multiplication in $\mathbb{Z}$ using the definition of the addition

• define the division

• define $\mathbb{Q}$, extend the definitions of the operations you have to $\mathbb{Q}$

• define the limit and the adherence space

• define $\mathbb{R}$ as $ \mathbb{\bar Q}$ (adherence of the rationals)

• find out some properties of $\mathbb{R}$

• extend the operations you have to $\mathbb{R}$

I hope this gives you a global picture on how some parts of mathematics (and the addition) are build