Set-Theoretic Foundations of Numbers and Functions

Set-Theoretic Foundations of Mathematics

It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets. This means notions such as natural numbers, integers, and real and rational numbers are defined in mathematics to be certain sets. Also, the very notion of a function is defined as a set.

Why is this done this way? A real number is not a set in our real life, so why do mathematicians define it in terms of sets? To understand this, one must go back to the origins of mathematics.

One of the first successful theories of mathematics was Greek geometry. The Greeks thought, however, that every number is either natural or rational. The discovery that $\sqrt{2}$ is not a fraction of two natural numbers was very disturbing to the Greeks. This caused a division between algebra and geometry. Algebra was seen by the Greek mathematicians to be unreliable and to lead to paradoxes. Geometry, however, was considered pure and was based on certain axioms that were seen as true statements about the real world.

It took a long time — until the 19th century — for mathematicians to realize that there were other geometries. Hyperbolic geometry was very similar to Greek Euclidean geometry, but also had important differences. It was realized that perhaps space did not behave according to Euclidean geometry (a fact that was confirmed later by Einstein’s theory of relativity). So what was the justification then for Euclid’s axioms? We cannot say anymore that they are obvious statements about the real world. The alternative is that they are a system invented by the mind.

If Euclidean geometry is invented by the mind, then how are we so sure it is consistent? How are we sure that we can accept these axioms and that no paradox would arise? We cannot. Thus even the very pure geometry was seen to be potentially flawed, and with it every result in Euclidean geometry was suspect because its axioms might be paradoxical.

The idea of the 19th century was to base mathematics on something different than geometry: sets. It was seen that the real numbers could be constructed out of rational numbers. So if the rational numbers form a consistent system, then there is no problem for the reals either. The rational numbers can be constructed out of the integers, and the integers can be constructed out of natural numbers. Thus, if the natural numbers are consistent, then so are the integers and the rationals.

The consistency of the natural numbers was more difficult, but it was seen to be possible to define natural numbers using only sets. Consequently, if the entire theory of sets is consistent, then so are the natural numbers.

And this is where the good news ends. Gödel’s incompleteness proof showed that it is impossible to demonstrate the consistency of set theory from within set theory itself. Nevertheless, most mathematicians do not doubt that set theory is consistent; they are simply unable to prove it.

What should we make of all those construction results? They are important because defining real numbers or natural numbers in terms of sets means we have to take less for granted. If we did not have those definitions in terms of sets, then outside of set theory we would also need a separate consistent theory of real numbers. Defining reals in terms of sets makes our lives easier in that sense.

Note that the construction does not tell us what real numbers are. A real number such as $\sqrt{2}$ is not a set in our minds, since we like to think of it as a point on a real number line. All that the construction of $\mathbb{R}$ tells us is that we can at least model something that looks like $\mathbb{R}$ inside set theory. Whether that something is $\mathbb{R}$ or not is a philosophically difficult question. But mathematicians do not care: they care only about the properties of an object. So if the set-theoretic model of $\mathbb{R}$ behaves exactly like $\mathbb{R}$, then it is $\mathbb{R}$ to a mathematician.

Functions as Sets

It is useful to explain how functions are modeled in set theory. First, we begin with the ordered pair $(a,b)$. This ordered pair is defined as

$$ (a,b)=\{\{a\},\{a,b\}\} $$

This definition satisfies the important property $(a,b) = (a’,b’)$ if and only if $a=a’$ and $b=b’$ (prove it!). Again, this is enough for the mathematician, since that property tells us exactly how the object behaves. A mathematician does not care whether the ordered pair is literally the thing we have defined above; the mathematician only cares that it has the same properties as the ordered pair, and it does.

It is then possible to define ordered $n$-tuples by setting $(a,b,c) = ((a,b),c)$ and $(a,b,c,d) = ((a,b,c),d)$ and so on. This again behaves as we expect.

A function $f$ between sets $A$ and $B$ is denoted $f:A\rightarrow B$. It is defined as the $3$-tuple $(A,B,R)$, where $R$ is a subset of $A\times B= \{(a,b)~\vert~a\in A,b\in B\}$. For this to be a function it must satisfy the condition that for each $a\in A$ there is a unique $b\in B$ such that $(a,b)\in R$.

For example, take the function $f:\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow x^2$. This function is formally given by

$$ (\mathbb{R},\mathbb{R},\{(x,x^2)~\vert~x\in \mathbb{R}\}) $$

So we see a function as “given” if we know its domain, its codomain and its graph. Indeed, if we use the shorthand $b = f(a)$ instead of $(a,b)\in R$, then two functions $f = (A,B,R)$ and $g = (A’,B’,R’)$ are equal if and only if $A=A’$, $B=B’$ and for each $a\in A$ we have $f(a)=g(a)$. This result matches the goal of our definition.

Notice that a function is intuitively seen as a kind of “process” or “change” in the sense that a function transforms an input to an output. In mathematics, a function is just a set that relates input variables to output variables. It is important to realize that the mathematical definition does not tell us what a set is; what a set is depends on your philosophical view. The mathematical definition merely tells us how to model the intuitive “function concept” in terms of set theory only. This model behaves the way we want it to behave, and the mathematician desires nothing more than that.