FOM: Dedekind --- The ultimate definition of cut?
pratt at cs.Stanford.EDU
Fri Mar 13 21:01:03 EST 1998
>The fact that the complement of the cuts would work just as well as the
>cuts themselves as representations of the reals should make it clear
>that it is meaningless to identify the reals with the cuts.
>This everything is meaningless because the
>"complements of cuts" is a meaningless notion.
>Indeed, a CUT (or: Dedekind cut) is a partition
>of the rationals onto two disjoint sets A and B,
>such and such ... .
That's very clever.
>Dedekind was very clever. He knew that prominent
>professors would ask why he had taken A rather than B
>to represent a real, or vice versa. So he took the
>pair of A and B, leaving for the nowadays categorists
>to be kidding over their own interpretations.
No, I would say that it is you that are clever. If you look closely at
Dedekind's definition you will see that he defines a cut ("Schnitt") not
as a partition but as "any separation of the system R into two classes
A1, A2 [such that] every number a1 in A1 is less than every number a2
This definition is entirely equivalent to choosing A1 as the representing
subset. With it, "complement" is the same thing as replacing "less than"
by "greater than". (This glosses over the as yet unaddressed issue of
cuts exactly at rationals.)
Your definition of cut is better than Dedekind's because as you observe it
avoids this arbitrariness. A partition of a set is properly understood
as an equivalence relation on that set. Here the above arbitrariness
disappears. One can then define a cut as a two-block equivalence relation
(i) (Each block is convex) If x and z are equivalent and y is between
x and z, then y is equivalent to x (and hence to z);
(ii) (At most two blocks) At least two of x, y, and z are equivalent; and
(iii) (At least two blocks) There exist x and y not equivalent to
(Dropping (iii) yields the projective real line, which has only one point
at infinity. With Dedekind's definition you get two points at infinity,
so Dedekind's notion of cut is not projective in that sense.)
But there still remains one small residue of arbitrariness here.
Although each irrational determines a unique cut, each rational
determines two cuts, which block of the partition does it go in?
Dedekind addressed this by identifying the two cuts (which I feel is
mathematically preferable to the common tactic of merely excluding one).
But the price for the nonarbitrariness of that solution is that a real
becomes a more complicated object than a mere partition.
But we live in modern times with such modern conveniences as the
partial equivalence relation (PER), defined as a symmetric transitive
binary relation. (The absence of reflexivity permits elements that
are not equivalent to themselves. By symmetry and transitivity such
elements cannot be equivalent to any other element either and hence may
be understood as undefined elements unrelated to anything.)
We can then define a real (or at least a cut) as follows.
* A real is a PER on the rationals consisting *
* of exactly two nonempty convex open blocks. *
(I defined convex above. By open I mean that for all y in a block
there exist x and z in that block such that x<y<z. In the standard
terminology of equivalence relations qua partitions, "nonempty" is
redundant, I included it only for clarity.)
At most one rational can be undefined or we would violate either the
two-block requirement or convexity. And openness forces those cuts
denoting rationals to have that rational be undefined. All other cuts
contain no undefined rational and denote irrationals.
This definition captures the underlying intuition of Dedekind's cut even
better than an ordinary partition. For when the grim reaper's scythe
slices through an irrational point it cleaves the rationals neatly into
two blocks. But when an unsuspecting rational receives the mortal blow
head-on so to speak, it does not equivocate over which side to escape to,
but simply exits with stoic grace. The beauty of the above definition
that this case is handled without mess behind the scenes.
(As a passing remark, PER's on a set X can always be coded by ER's on
"the" set X+1 having one new element, collecting all the undefined
elements into a block that includes the new element as a marker for the
"undefined" block. The PER's above can be coded by ER's without any
additional elements, with the irrationals being two-block ER's and the
rationals being three-block ER's one block of which is a singleton.
However I do not see how to axiomatize such ER's as neatly as was
possible above with PER's.)
The definition of a real as the set of rationals strictly less than it
does not make explicit provision for those reals that are rationals, but
arbitrarily chooses "less than" over the equally good "greater than".
Kanovei's vision of a cut as a partition, realized more precisely as
a partial equivalence relation, makes neither special provision for
those reals that are rational nor arbitrary choices between equally
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