Reconciling my understanding of real numbers as Dedekind cuts with Dedekind and Penrose

Question

Penrose (The Road to Reality, Section 3.2) describes Dedekind as defining real numbers via a "knife-edge" cut in the size-ordered sequence of rationals, separating them into two sets; where a cut does not fall on a rational, it must therefore fall on a kind of number which we call real.

I find two flaws in this. In what follows I shall use the term "two sets of rationals" to indicate a division of all rationals into two sets, ordered by size.

The first flaw appears to be circular reasoning. On what basis does Dedekind allow that a gap between the two sets of rationals has any meaning? It seems to me that he implicitly assumes the existence of reals, for which he must already have a definition in order to assume anything about them. Assuming your result is not valid logic.

Next, consider a convergent series and its termination, such as 1/2 + 1/4 + 1/8 ... = 2. The shortfall in the series gets smaller and smaller (in this example, at any point in the series it is equal to the latest term in the series). Yet, in the calculus at least, we are happy to accept that in the limit, the shortfall vanishes and the equality expressed above is valid; this is how we justify a continuous derivative of a continuous curve, for example. Now apply this to the rational number line. As more rationals are included, the gaps between the two sets of rationals shrinks. In the limit, by the reasoning of the calculus the gap also vanishes, as do all gaps where other cuts might fall, and the rational number line becomes continuous. This appears to undermine the assumption of a gap between two sets of rationals.

But I am somewhat confused by this question on Proof on density of rationals and irrationals in R via dedekind cut, which appears to treat a Dedekind cut as a much blunter instrument than Penrose does, by talking of "infinitely many rationals that are greater than other rational in our dedekind cut".

Who offers the flawed argument; Dedekind, Penrose or me, and where is the flaw?

Answer 1

The first flaw appears to be circular reasoning. On what basis does Dedekind allow that a gap between the two sets of rationals has any meaning?

It only relies on the existence of the total order on rational numbers. If you have two nonempty sets $A,B$ of rationals with the property that $a<b$ for all $a\in A$ and $b\in B$, and $A\cup B=\mathbb Q$, this partition of the rationals has a suggestive "void" between them, and those are the Dedekind cuts that contribute new nonrational points.

Now apply this to the rational number line. As more rationals are included, the gaps between the two sets of rationals shrinks. In the limit, by the reasoning of the calculus the gap also vanishes, as do all gaps where other cuts might fall, and the rational number line becomes continuous.

I would accept this as a heuristic explanation of why the addition of Dedekind cuts fill up the gaps in the rational number line, but you're speaking as if it were a justification that the rational number line is already continuous. It isn't, at least it isn't complete in the ways we want it to be. There is an increasing sequence of rationals converging to $\sqrt{2}$, but the limit is not in $\mathbb Q$ and that is the point. There's a hole where the limit should be. Indeed, calculus with rationals isn't worth doing until you've filled up these holes with real numbers.

It sounds a little like you are confusing density for completeness. The first one means points get in all the open sets, but the second means that the set is maximally packed, in a sense.

Certainly there aren't any intervals of positive length between rational numbers. (Perhaps that is what you assumed a gap would be.) But that reflects their density not their completeness. The irrationals are also dense, but somehow the two sets, while disjoint, intermingle in the entirety of the real line.

Who offers the flawed argument; Dedekind, Penrose or me, and where is the flaw?

There are no flaws in Dedekind's argument and as far as I can tell Penrose's account is an accurate description of Dedekind cuts. By process of elimination, it seems the issue is currently with your understanding of the matter.

Please continue to clarify what exactly you see is an important difference. Perhaps the most important ones were already dealt with above, but if not, please provide some more concrete definitions and things you see as opposing each other. I can tell something's bugging you, but until you formulate what it is exactly, it'll be hard to help any more.

Answer 2

Over and over you say that if we assume X then things are inconsistent or circular, when in fact nobody ever assumed X! I bet I know what happened; I bet you're confusing the "motivation" with the actual official definition-theorem-proof exposition.

Huh? For example, if I were writing about Dedekind cuts I'd probably start here:

(*) For each $x\in\Bbb R$ define $S_x=\{r\in\Bbb Q: r>x\}$. It's easy to see that the mapping $x\mapsto S_x$ is injective. In fact knowing $S_x$ tells us everything there is to know about $x$, so we may as well go ahead and define the reals in terms of the sort of sets that arise as S_x.

Then I'd go ahead and define Dedekind cuts and prove that they do what they need to do. What's important to realize is that (*) above is NOT part of the "official" math. If (*) were included then yes, things would be circular. But officially (*) has nothing to do with anything; it's just "motivation", trying to see how someone might have come up with that definition. (In particular you will never see the notation "$S_x$" in the official part...)