Question on cut

Question

I was reading principle of mathematical analysis by Walter Rudin and there i saw this definition of cut: Consider a set $a\subset Q$ to be a cut. If it follows the following property

(i)$a$ is non empty. And $a\not=Q$

(ii)if $p\in a $, $q\in Q $ with $q <p$. Then $q\in a $.

(iii) $p \in a$ then $p <r $ for some $r\in a $.

So basically a cut cannot have largest number. But I was then studying An essay on theory of numbers by R. Dedekind for further study.There it is written that if a rational number $r $ produces cut then the number can belong to the first set. That is the set in which all elements $\leq r $. So it can have largest element. So which one is true?? Any help would be appriciated. Thanks.

Answer 1

Dedekind's "Essays on the Theory of Numbers" is now available on Project Gutenberg. On page 6 he defines

If now any separation of the system $R$ into two classes $A_1, A_2$ is given which possesses only this characteristic property that every number $a_1$ in $A_1$ is less than every number $a_2$ in $A_2$, then for brevity we shall call such a separation a cut [Schnitt] and designate it by $(A_1,A_2)$.

($R$ is the set of rational numbers, $\Bbb Q$ in today's convention.)

You will note that for a rational number $r$, he allows both $$(\{ a\in \Bbb Q\mid a < r\}, \{ a\in \Bbb Q\mid r \le a\})$$ and $$(\{ a\in \Bbb Q\mid a \le r\}, \{ a\in \Bbb Q\mid r < a\})$$ as cuts. This results in required "understandings" such as is mentioned in the next line:

We can then say that every rational number $a$ produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different;

In modern sensibilities, there are a few changes that are made to his approach. One is that in his definition, he does not explicitly require that neither $A_1$ nor $A_2$ should be empty. To his conception, this was just understood from the wording. Now we always make this requirement explicit (without it, you end up with the extended Reals instead of just the Reals).

The final change is that we don't like having to "not look upon" certain pairs of cuts "as essentially different". Instead we prefer to change his definition so that each rational number produces only one cut, not two. How to do this differs from author to author. Most commonly, we now require the lower set to have no highest element. Also, since the lower set uniquely determines the higher set, many authors, including Rudin, abandon the higher set altogether and just use the lower set to define the cut.

However, these choices are not universal. For example, this writer chooses to drop the requirement that $A_1\cup A_2 = \Bbb Q$, but adds some additional restrictions that essentially means that $A_1\cup A_2$ is all of $\Bbb Q$ except for possibly one point. In his cuts, neither $A_1$ nor $A_2$ ever has an extremum, and in the cut produced by a rational number, that number is the one point skipped.