Ramble, Part 63: Incisive Thinking
Nov. 26th, 2008 12:33 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowMost of Richard Dedekind's contributions to mathematics were in the area of algebra; one of the fundamental constructs used in algebraic number theory1 is the "Dedekind domain". But he also supplied an alternative construction of the real numbers from the rationals, which will be detailed under the cut.
Dedekind's construction, like Cauchy's, can be approached via the standard decimal representation. To say that Sqrt[2]=1.4142... is to say, among other things, that Sqrt[2] is greater than 1, 1.4, 1.41, etc., and less than 2, 1.5, 1.42, and so on. A real number, that is, can be identified by trapping it between two sets of rational numbers - one consisting of those less than the given real, and one consisting of those greater. As with Cauchy's construction, the difficulty lies in speaking of these two sets without identifying beforehand the real number that separates them.
Dedekind's answer was to define what is now called a "Dedekind cut". This consists of a pair of nonempty sets of rational numbers (L,U) with the following properties: every rational number belongs to either L or U; no rational number belongs to both; and every rational number in L is less than every rational number in U. Dedekind declares that a real number is simply a Dedekind cut - not "is defined by"; "is". We'll call L the "lower set" of the cut and U the "upper set".
There's one small problem, easily dealt with. An irrational number corresponds to only one cut, but a rational number r corresponds to two, depending on whether r itself is assigned to L or to U. (It must be assigned to one or the other!) One could, for instance, arbitrarily require that U not have a least element; this amounts to across-the-board assignment of r to L.
It is easy to define addition of two Dedekind cuts (L,U) and (L',U'). If we use l, l' to represent typical elements of L, L' respectively, and likewise for U, U', then the lower set for (L,U)+(L',U') has typical elements of the form l+l', and the upper set has u+u'. Multiplication is not so easy, because multiplication does not respect order the way addition does. (If l < u and l' < u', then l+l' < u+u', but it's not necessarily true that ll' < uu'; let l = l' = -1, u = u' = 0.) If 0 is in both L and L', then the upper set of the product has typical elements uu', and all other rational numbers lie in the lower set. To deal with other cases, define -(L,U) to be (-U,-L), and use - as assumptions - the equalities (-a)b=-(ab), (-a)(-b)=ab (where a,b represent two Dedekind cuts).
It's interesting to note that Dedekind cuts bear a strong resemblance to Eudoxus's definition of proportionality.
Although Cauchy sequences and Dedekind cuts are quite different in construction, it's fairly easy to show that the notions of "real number" obtained in these two ways are equivalent. Either way, the question of the nature of the real numbers has been reduced to questions concerning the rational numbers, which are much more tractable.
Or so it was thought; we'll see more on that point later. Dedekind will reappear in that story. First, though, we need to take a detour involving a fellow named Cantor, whose interest in Fourier series led him on an unexpected tangent.
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1. As opposed to elementary number theory2 and analytic number theory.
2. Despite its name, this is an independent subfield, with techniques and subtleties all its own.
Ramble Contents
Dedekind's construction, like Cauchy's, can be approached via the standard decimal representation. To say that Sqrt[2]=1.4142... is to say, among other things, that Sqrt[2] is greater than 1, 1.4, 1.41, etc., and less than 2, 1.5, 1.42, and so on. A real number, that is, can be identified by trapping it between two sets of rational numbers - one consisting of those less than the given real, and one consisting of those greater. As with Cauchy's construction, the difficulty lies in speaking of these two sets without identifying beforehand the real number that separates them.
Dedekind's answer was to define what is now called a "Dedekind cut". This consists of a pair of nonempty sets of rational numbers (L,U) with the following properties: every rational number belongs to either L or U; no rational number belongs to both; and every rational number in L is less than every rational number in U. Dedekind declares that a real number is simply a Dedekind cut - not "is defined by"; "is". We'll call L the "lower set" of the cut and U the "upper set".
There's one small problem, easily dealt with. An irrational number corresponds to only one cut, but a rational number r corresponds to two, depending on whether r itself is assigned to L or to U. (It must be assigned to one or the other!) One could, for instance, arbitrarily require that U not have a least element; this amounts to across-the-board assignment of r to L.
It is easy to define addition of two Dedekind cuts (L,U) and (L',U'). If we use l, l' to represent typical elements of L, L' respectively, and likewise for U, U', then the lower set for (L,U)+(L',U') has typical elements of the form l+l', and the upper set has u+u'. Multiplication is not so easy, because multiplication does not respect order the way addition does. (If l < u and l' < u', then l+l' < u+u', but it's not necessarily true that ll' < uu'; let l = l' = -1, u = u' = 0.) If 0 is in both L and L', then the upper set of the product has typical elements uu', and all other rational numbers lie in the lower set. To deal with other cases, define -(L,U) to be (-U,-L), and use - as assumptions - the equalities (-a)b=-(ab), (-a)(-b)=ab (where a,b represent two Dedekind cuts).
It's interesting to note that Dedekind cuts bear a strong resemblance to Eudoxus's definition of proportionality.
Although Cauchy sequences and Dedekind cuts are quite different in construction, it's fairly easy to show that the notions of "real number" obtained in these two ways are equivalent. Either way, the question of the nature of the real numbers has been reduced to questions concerning the rational numbers, which are much more tractable.
Or so it was thought; we'll see more on that point later. Dedekind will reappear in that story. First, though, we need to take a detour involving a fellow named Cantor, whose interest in Fourier series led him on an unexpected tangent.
Previous Next
1. As opposed to elementary number theory2 and analytic number theory.
2. Despite its name, this is an independent subfield, with techniques and subtleties all its own.
Ramble Contents
