Real number (as Dedekind cut)

com­ment: Mnemon­ics for defined macros: \Ql = Q left, \Qr = Q right, \Qls = Q left strict, \Qrs = Q right strict.

The ra­tio­nal num­bers have a prob­lem that makes them un­suit­able for use in calcu­lus — they have “gaps” in them. This may not be ob­vi­ous or even make sense at first, be­cause be­tween any two ra­tio­nal num­bers you can always find in­finitely many other ra­tio­nal num­bers. How could there be gaps in a set like that? \(\newcommand{\rats}{\mathbb{Q}} \newcommand{\Ql}{\rats^\le} \newcommand{\Qr}{\rats^\ge} \newcommand{\Qls}{\rats^<} \newcommand{\Qrs}{\rats^>}\) \(\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\sothat}{\ |\ }\)

But us­ing the con­struc­tion of Dedekind cuts, we can suss out these gaps into plain view. A Dedekind cut of a to­tally or­dered set \(S\) is a pair of sets \((A, B)\) such that:

  1. Every el­e­ment of \(S\) is in ex­actly one of \(A\) or \(B\). (That is, \((A, B)\) is a par­ti­tion of \(S\).)

  2. Every el­e­ment of \(A\) is less than ev­ery el­e­ment of \(B\).

  3. Nei­ther \(A\) nor \(B\) is empty. (We’ll see why this re­stric­tion mat­ters in a mo­ment.)

One ex­am­ple of such a cut might be the set where \(A\) is the nega­tive ra­tio­nal num­bers and \(B\) is the non­nega­tive ra­tio­nal num­bers (pos­i­tive or zero). We see that it satis­fies the three prop­er­ties of a Dedekind cut:

  1. Every ra­tio­nal num­ber is ei­ther nega­tive or non­nega­tive, but not both.

  2. Every ra­tio­nal num­ber which is nega­tive is less than a ra­tio­nal num­ber that is non­nega­tive.

  3. There ex­ists at least one nega­tive ra­tio­nal num­ber (e.g. \(-1\)) and one non­nega­tive ra­tio­nal num­ber (e.g. \(1\)).

In fact, Dedekind cuts are in­tended to rep­re­sent sets of ra­tio­nal num­bers that are less than or greater than a spe­cific real num­ber (once we’ve defined them). To rep­re­sent this, let’s call them \(\Ql\) and \(\Qr\).

Know­ing this, why does it mat­ter that nei­ther set in a Dedekind cut is empty?

If \(\Ql\) were empty, then we’d have a real num­ber less than all the ra­tio­nal num­bers, which is \(-\infty\), which we don’t want to define as a real num­ber. Similarly, if \(\Qr\) were empty, then we’d get \(+\infty\).

Com­ple­tion of a space

If a space is com­plete (doesn’t have any gaps in it), then in any Dedekind cut \((\Ql, \Qr)\), ei­ther \(\Ql\) will have a great­est el­e­ment or \(\Qr\) will have a least el­e­ment. (We can’t have both at the same time — why?)

Sup­pose \(\Ql\) had a great­est el­e­ment \(q_u\) and \(\Qr\) had a least el­e­ment \(q_v\). We can’t have \(q_u = q_v\), be­cause the same num­ber would be in both sets. So then be­cause the ra­tio­nal num­bers are a dense space, there must ex­ist a ra­tio­nal num­ber \(r\) so that \(q_u < r < q_v\). Then \(r\) is not in ei­ther \(\Ql\) or \(\Qr\), con­tra­dict­ing prop­erty 1 of a Dedekind cut.

But in the ra­tio­nal num­bers, we can find a Dedekind cut where nei­ther \(\Ql\) nor \(\Qr\) have a great­est or least el­e­ment re­spec­tively.

Con­sider the pair of sets \((\Ql, \Qr)\) where \(\Ql = \set{x \in \rats \mid x^3 \le 2}\) and \(\Qr = \set{x \in \rats \mid x^3 \ge 2}\).

  1. Every ra­tio­nal num­ber has a cube ei­ther greater than 2 or less than 2,

  2. Be­cause \(f(x) = x^3\) is a mono­ton­i­cally in­creas­ing func­tion, we have that \(p < q \iff p^3 < q^3\), which means that ev­ery el­e­ment in \(\Ql\) is less than ev­ery el­e­ment in \(\Qr\).

So \((\Ql, \Qr)\) is a Dedekind cut. How­ever, there is no ra­tio­nal num­ber whose cube is equal to \(2\), so \(\Ql\) has no great­est el­e­ment and \(\Qr\) has no least el­e­ment.

This rep­re­sents a gap in the num­bers, be­cause we can in­vent a new num­ber to place in that gap (in this case \(\sqrt[3]{2}\)), which is “be­tween” any two num­bers in \(\Ql\) and \(\Qr\).

Defi­ni­tion of real numbers

Be­fore we move on, we will define one more struc­ture that makes the con­struc­tion more el­e­gant. Define a one-sided Dedekind cut as any Dedekind cut \((\Ql, \Qr)\) with the ad­di­tional prop­erty that the set \(\Ql\) has no great­est el­e­ment (in which case we now call it \(\Qls\)). The case where \(\Ql\) has a great­est el­e­ment \(q_g\) can be triv­ially trans­formed into the equiv­a­lent case on the other side by mov­ing \(q_g\) to \(\Qr\) where it is au­to­mat­i­cally the least el­e­ment due to be­ing less than any other el­e­ment in \(\Qr\).

Then we define the real num­bers as the set of one-sided Dedekind cuts of the ra­tio­nal num­bers.

  • A ra­tio­nal num­ber \(r\) is mapped to it­self by the Dedekind cut where \(r\) it­self is the least el­e­ment of \(\Qr\). (If the cuts weren’t one-sided, \(r\) would also be mapped to the set where \(r\) was the great­est el­e­ment of \(\Ql\), which would make the map­ping non-unique.)

  • An ir­ra­tional num­ber \(q\) is newly defined by the Dedekind cut where all the el­e­ments of \(\Qls\) are less than \(q\) and all the el­e­ments of \(\Qr\) are (strictly) greater than \(q\).

Now we can define the to­tal or­der \(\le\) for two real num­bers \(a = (\Qls_a, \Qr_a)\) and \(b = (\Qls_b, \Qr_b)\) as fol­lows: \(a \le b\) when \(\Qls_a \subseteq \Qls_b\).

Us­ing this, we can show that un­like in the Cauchy se­quence defi­ni­tion, we don’t need to define any equiv­alence classes — ev­ery real num­ber is uniquely defined by a one-sided Dedekind cut.

If \(a = b\), then \(a \le b\) and \(b \le a\). By the defi­ni­tion of the or­der, we have that \(\Qls_a \subseteq \Qls_b\) and \(\Qls_b \subseteq \Qls_a\), which means that \(\Qls_a = \Qls_b\), which means that the Dedekind cuts cor­re­spond­ing to \(a\) and \(b\) are also equal.

Proof of the field struc­ture of Dedekind cuts.