Proper definition of addition and multiplication

Question

I recently started to study maths at university and in the analysis course we started, as usual, by looking at the axioms of $\mathbb R$ as a field. I think, I've understood the underlying intuition of these axioms quite well, but one question remained open for me:

How do we define, what we do when we add two numbers, i.e. how to really calculate the result. Is the intuitive addition the only interesting function from $\mathbb R^2$ to $\mathbb R$ which satisfies these axioms? Is this question even relevant?

I thought, I could give it a try, and made the following rules for addition and multiplication (which are more or less accurate):

(1) If $m,n\in\mathbb N_0$, then there exists to sets $E,F$ with cardinality $m$ and $n$ respectively for which $E\cap F=\{\}$. Then: $$ m+n:=|E\cup F| $$ Now define the inverses.

(2) Now either define $$ mn:=|\{(e,f):e\in E,\space f\in F\}| $$ or $$ mn:=\underbrace{m+m+...+m}_{n} $$ Then define the inverses.

From this we can extend the addition to the rationals by: $$ \frac ab +\frac cd:=\frac{ad+bc}{bd} $$ And even to the reals with cantors limit construction of $\mathbb R$.

The principle that I don't fully understand, is that we never seem to define our proper addition and multiplication. It seems to be always implicit what we mean by $+$ and $*$. How to explain this?

Answer 1

If you introduce $\mathbb R$ axiomatically, there is no urge to define $+$ or $\cdot$ or $42$ in a concrete fashion. You may take any other set than the "true" real numbers and any other operations than the "true" addition and multiplicatioan - as long as your choices still make the axioms of a complete ordered field hold, you can validly claim that the set you have in mind is "the" set of real numbers and that the operations you have in mind are "the" addition and multiplication of real numbers. The good thing is: Whenever you need want to communicate a fact about (your) real numbers to another mathematician, you must express numbers in a ways he/she is bound to understand correctly, such as

• $0$ and $1$ for the neutral elements of addition/multiplication,
• $42$ as the real number obtained from $(1+1)\cdot(1+1+1)\cdot(1+(1+1)\cdot(1+1+1))$,
• $\sqrt 2$ as the unique real number $w$ that has the property $w\cdot w=1+1$ and for which there exists real $u$ with $u\cdot u=w$,

and so on.

Answer 2

$\mathbb{R}$ is up to isomorphism the only ordered field with the supremum property, so we don't really need to know the definition of $+$ and $*$ to work with the arithmetic of $\mathbb{R}$.

Now, what we do want to know is whether such an objetc exists. To do so, we have to define what a real number is, what $+$, $*$ and $<$ are, to prove that there are inverses and so on.

There are several problems with your definitions:

-You implicitely assume that there are elements in $\mathbb{R}$ with which we can count the number of elements in a finite set: who are they?

-You implicitely assume that there are inverses for the additive law (for natural integers): who are they?

-You seem forget to say how to add whole integers that are not positive

-You implicitely assume there are inverses for the multiplicative law (for natural integers): who are they? (what is $\frac{1}{n}$?)

-etc...

There are many things to do in order to define theses laws, many proofs of existence. Defining the field of real numbers is more complicated and longer than just doing "elementary" real analysis and arithmetic. That is why we often rely on an axiomatic background such as the distributivity of $*$ over $+$ and other rules, and we forget about the rest.

If you are interested in the subject, you can learn about the construction of sets of numbers in mathematics. Discovering this can be quite a great experience.

My advice would be to start with the construction of $\mathbb{Z}$ using $\mathbb{N}$ (easy), then that of $\mathbb{Q}$ using $\mathbb{Z}$ (similar), then that of $\mathbb{R}$ using $\mathbb{Q}$ (more difficult and longer), then that of $\mathbb{N}$ using the axioms of set theory (more difficult and subtler).