Basic atoms in mathematics

Question

Given the concepts '1', 'set' and 'sum' (and maybe 'point' for geometry), can you build the whole mathematics upon then? If not, what other basic atoms would you need?

Answer 1

In principle, yes: let $\kappa$ denote a regular cardinal. Then the category $\mathbf{Set}_\kappa$ (of all $\kappa$-small sets) is the $\kappa$-cocomplete category freely generated by $\{1\}$. In plain English: $\kappa$-small sums and $1$ are together enough to be build every set and every function.

But in practice, the answer is no: for one, what in the world is a category, what's a regular cardinal, and how do we reason about these things? The whole things ends up being circular. To prevent circularity, we need to write down some axioms that explain how to reason either about sets, or else about something equally as fundamental like categories or $\infty$-groupoids, or something like that.

There's an important lesson here, which is that axiomatizing a structure is typically a lot harder than merely defining it. In fact, the Godel incompleteness theorems implies that any time we're trying to axiomatize a structure that contains (as a definable substructure) the set of natural numbers $\mathbb{N}$, together with the element $1 \in \mathbb{N}$ and the functions $+$ and $\times$ on $\mathbb{N}$, we're going to run into problems.

Answer 2

You might want to have a look at Peano arithmetic. That is a fairly minimal set of rules that can be used to create numbers.

For natural numbers you pretty much need only your starting point (0), an operation (+1) and to permit infinite repetition (of +1) - this is alternatively defined in set theory as the law of the unique successor.

To then get the negative integers, it is sufficient to say that your +1 operation has an inverse (-1).

Multiplication actually follows from repetition of addition, and this is sufficient to give it its distributive, commutative and associative properties but in algebra it is usually specified as an additional operation. But this still only allows you to reach whole numbers.

Now, if you want to create all the rational numbers (including fractions), it is sufficient to say that your multiplication has an inverse function (which you know as division) and from these rules you can create all rational fractions.

Then to create irrational numbers you need to state what it means to take a power of something - namely exponentiation. If you exponentiate with only whole numbers, this does not extend your numbers but if you allow fractions this implies that exponentiation has an inverse function, and this is sufficient to create the irrational numbers such as $\sqrt{2}$. If we permit a square root of a negative number, this also creates imaginary numbers denoted $i$.

Finally, if you allow an infinite sum of some series provided it converges, this will give transcendental numbers such as $e$ and $\pi$.

These are all of the complex numbers denoted $\mathbb{C}$ but you can go still further with this, in rare circumstances.

So to sum up, to create the numbers we mostly work with you only really need 0, 1, the addition function, multiplication, and every function having an inverse function.

Answer 3

Your question raises the issue: What does it mean for a concept to be "given"?

Certainly it does not just mean that the symbol itself is given, e.g. the symbols '1', 'set', ... as in your example.

In addition, for a concept to be given, one must also state the rules, or behaviors, or postulates, or (yes) axioms which govern how those symbols are to be manipulated.

As an example, let's look at set theory. Certainly 'set' is one of the givens (although I shall back off of that in just a minute). But 'set' all on its own just isn't enough. One also needs the 'element of' relation: without that, the concept of 'set' is incomplete because it is impossible to talk about the elements of that set. With the 'element of' relation, one can form some interesting sentences about set theory:

1. For all sets $A$ and $B$, if it is true that for each set $C$, $C$ is an element of $A$ if and only if $C$ is an element of $B$, then $A=B$.
2. There exists a set $A$ such that for all sets $B$, $B$ is not an element of $A$.

The first statement is the axiom of extensionality, which says that a set is determined by its elements. The second statement is the axiom of the empty set, which asserts the existence of the empty set. Those are two of the ZFC axioms for set theory; there's still a bunch more.

So, if for a concept to be "given" means that certain 'atoms' as you call them are given, and certain axioms regarding those atoms are also stated, then (according to current understanding) all mathematics can indeed be built up in this manner. For example:

1. The ZFC axioms for set theory are regarded as a basis for mathematics. In the universe of formal ZFC it is not actually necessary to introduce the terminology 'set', since intuitively every object is a set. However, the 'element' relation is of key importance.
2. Peano's axioms for arithmetic are regarded as a basis for some parts of modern number theory and analysis. In Peano arithmetic one indeed uses '1' to represent the smallest natural number, but that's not enough. One needs the 'successor function' to represent the function which given a natural number $x$ outputs its successor, which with our advanced understanding is eventually seen to be $x+1$.
3. Euclidean geometry remains a basis for a certain limited part of modern geometry. One indeed uses 'point'. But one also needs 'line', and one needs a relation of 'incidence' where a sentence like "a point is incident to a line" means that (from a modern, set theoretic perspective) the point is an element of the line. And that's still not enough. One also needs a 'betweenness' relation amongst the points that are incident to a given line. The exact details were worked out by Hilbert and are presented in a book by Hartshorne.

Finally, what about the collection of concepts that you originally asked about, '1', 'set', 'sum', 'point'? They are plucked from different parts of mathematics, and most likely are not enough to cover any one portion of mathematics, even if your question had laid out a bunch of proposed axioms as would be necessary for it to make full sense.

As said, 'set' with the 'element of' relation added, and with all the ZFC axioms carefully laid out, are currently regarded as enough to cover all mathematics.