Sets as numbers through Dedekind's cuts

Question

I am having some issues understanding the definition of real numbers as a Dedekind Cut. Firstly, a Dedekind cut is a set, not a number (namely, a set in which all its values are less than the value of the rational number q), so i don't see how can we proceed from that. Then, I don"t see the intuition behind defining real numbers as Dedekind, even as a starting point,but I think it comes from this first problem.

Just to make it clear, I don't see how $\pi$ is the set of all rational numbers smaller than $\pi$.

Answer 1

So I think a lot of your confusion stems from understanding how/why mathematics is formalised. Set theory is a very simple theory, from which we expect there to be no contradictions. Here's a lightly anachronistic account of why we may want this:

In 1903, Frege attempted to define arithmetic from simple laws; one of which was `Basic Law 5'. In a (rough) sense, he proposed that a set was defined by a property shared amongst its elements; a seemingly simple system. Bertrand Russel (see Russel's paradox) demonstrated that this actually leads to a contradiction, namely if we consider the set of all sets that do not contain themselves $X=\{x:x\notin x\}$. From there we just ask if $X\in X$. Either possibility leads to a contradiction.

So what mathematicians developed were various laws of sets, axioms (which we predict by raw intuition, but can never prove) that don't lead to contradiction. One popular one is Zermelo-Fraenkel (ZF) set theory. Here's the rub; since mathematics with more exotic ideas, such as real numbers, aren't immediately obviously not-self contradictory, by embedding them as sets, bound by the rules of (again for example) ZF set theory, we have a stronger reason to think that they aren't contradictory. That is why we embed them as sets, and define our rules on them using set theory (but not unless you explicitly do so; usually we accept that addition, as an example, just works).

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So in answer to the OP, it doesn't really matter how we define the real numbers, so long as it is defined within our set theory, and that we can prove the same things with the same definitions. Of course there are always aesthetic arguments over why one definition might be better than another, but from a general perspective, most do not really care.

Answer 2

One of the most foundational ideas in mathematics is the idea that if you can identify two things that behave in exactly the same way under every relevant context, then you can treat them as the same thing.

For example, the Peano axioms tell us that if we have a set that follows certain rules, then that set is equivalent to the natural numbers. You might see a particular construction of them as a sequence of sets looking something like:

$$\emptyset \\ \{\emptyset\} \\ \{\emptyset, \{\emptyset\}\} \\ \dots$$

such that the thing we normally think of as the number 2 is actually a set containing two elements. It takes some work to define operations like addition and multiplication in terms of set operations, but once you do so you've got a consistent arithmetic that is indistinguishable from the natural numbers we know and love.

To get the integers and rational numbers, we build sets of equivalence classes of natural numbers - for example, the integer $-5$ is actually a set of ordered pairs of the form $(n, 5 + n)$, and the fraction $\frac{17}{3}$ is a set of pairs $(17n, 3n)$, and we extend all of our operations from the natural numbers onto these sets. Notice that in these constructions there is a mapping of the natural numbers onto their equivalent representations - e.g. $5 \sim \{(5, 0), (6, 1), (7, 2), \ldots\} \subset \mathbb{Z}$, and $5 \sim \{(5, 1), (10, 2), (15, 3), \ldots\} \subset \mathbb{Q}$.

We can then build the real numbers in a similar way from the rationals. If we do it via Dedekind cuts, then we're looking at certain partitions of the rational numbers - specifically, we have partitions $\mathbb{Q} = A \cup B$ where all of $A$ is less than all of $B$, and $A$ is bounded from above but does not contain a maximal element. Any time $B$ does contain a minimal element, we label that partition with that value. So then we have to create operations on this set of partitions, preferably in a way so that if we only look at the partitions that we've mapped to rational numbers we get results consistent with similar operations on the original numbers (e.g. if we have $p \sim \{A_p | B_p\}$ and $q \sim \{A_q | B_q\}$, we want to define addition on the Dedekind cuts so that $p + q \sim \{A_{p+q} | B_{p+q}\}$).

The rationale or intuition behind these constructions is based in the idea of "completeness". We start with some construction that works for some, but not all, of our existing numbers, and we build a system that plugs the gaps. For example:

• Equations of the form $n + p = q$ sometimes don't have solutions in the natural numbers (when $p > q$), so we extend them to the integers. Similarly we need rational numbers to solve almost all equations like $pn = q$.

• In the natural numbers and the integers, any set that is bounded from above has a unique least upper bound, but this property vanishes from the rationals. The real numbers, via Dedekind cuts, restore it.

• If we consider Cauchy sequences (which you can think of as a kind of convergent sequence), then some sequences of rational numbers don't have a limit. So an alternative construction of the real numbers is based on these, and then you can have fun proving that this is consistent with Dedekind cuts.