Dedekind cuts & infinitesimals

[Rasalhague]

"A real number is a Dedekind cut in the set Q of rational numbers: a partition of Q into a pair of nonempty disjoint subsets <L,U> with every element of L less than every element of U and L having no largest member. Thus 2^1/2 can be identified with the cut: L = {q in Q: q² < 2}, U = {q in Q: q² > 2" (Goldblatt: Lectures on the Hyperreals, p. 12).

"A number d in an ordered field is called infinitesimal if it satisfies 1/2 > 1/3 > ... > 1/m > |d| for any natural counting number m = 1,2,3..." (Stroyan: Foundations of Infinitesimal Calculus, p. 9).

How can we tell that no more than one real number is defined by a Dedekind cut; is the answer that real numbers are simply defined as equal if they can be represented by a single Dedekind cut?

I don't understand Stroyan's proof, on p. 10, that there are no positive infinitesimals in the reals. He uses a Dedekind cut (A,B), where A = {a < 1/m: m = 1,2,3...} and B = {1/m < b: m = 1,2,3...}, then claims that "zero is at the gap in the reals and every positive real number is in B." But it seems to me that zero is in A rather than the gap. His cut seems to disprove the statement that there are no positive infinitesimals by defining one. What am I missing here?

 

[Hurkyl]

(note that the two books define "Dedekind cut" slightly differently)

There are three cases for a Dedekind cut in Stroyan's convention:

1. A has a maximum element. B does not have a minimum element.
2. B has a minimum element. A does not have a maximum element.
3. Neither A has a maximum nor B has a minimum.

The distinguishing property of the real numbers is that case #3 never happens for them. This property (together with being an ordered field) is often taken as the definition of a "field of real numbers".



Goldblatt is constructing a specific model of the field of real numbers, naming them by cuts of the rationals. He has chosen his definition of Dedekind cut in a carefully restricted way so that each cut defines a different real number. If you carried out the same construction using the set of all cuts by Stroyan's definition, you would have to impose the equivalence relation that cuts of type #1 and of type #2 both define the same real number, if the maximum of the A of the first is the minimum of the B of the second.

 

[gerben]

How can we tell that no more than one real number is defined by a Dedekind cut; is the answer that real numbers are simply defined as equal if they can be represented by a single Dedekind cut?

A Dedekind cut is performed on a totally ordered set.
Since L and U are, respectively, defined as being composed of elements smaller-than and larger-than "something" there cannot be two elements at the cut. If there were two elements, one of them would be smaller (or larger) than "something" since only one of them could be equal to "something".

I don't understand Stroyan's proof, on p. 10, that there are no positive infinitesimals in the reals. He uses a Dedekind cut (A,B), where A = {a < 1/m: m = 1,2,3...} and B = {1/m < b: m = 1,2,3...}, then claims that "zero is at the gap in the reals and every positive real number is in B." But it seems to me that zero is in A rather than the gap. His cut seems to disprove the statement that there are no positive infinitesimals by defining one. What am I missing here?

I think he just means there is always a fraction smaller than the smallest real.
Take $S = \{1/m: M \in \mathbb{N}\}$, then he defines sets A and B as follows:
- elements of A are smaller than all elements in S
- for every element of B there is an element of S that is smaller

So, there is always an element of S smaller than all elements of B; always a fraction smaller than the smallest real.

 

[Rasalhague]

In Goldblatt's definition, Dedekind cuts can only be performed on the rationals, and always define a real number. I see Stroyan's Dedekind cuts can be made in any "ordered field" (definition 1.4, p. 8). His definition doesn't explicitly rule out the possibility that A has a maximum and B a minimum, or does it? But perhaps that possibility is ruled out in the case of the particular fields he discusses here by the nature of these fields.

I'm still very puzzled by this. He says the sets A and B on p. 10 define a cut in "the rationals, reals, and hyperreal numbers." When he says "zero is at the gap in the reals", does he mean zero is the maximum of A when the cut is made on the reals? But that seems paradoxical, since when the cut defined by the same sets is made on the rationals, it defines/creates a real number greater than 0 and less than all numbers of the form 1/m where m = 1,2,3..., according to my (presumably flawed) interpretation of the method for defining the real numbers in terms of Dedekind cuts.

 

[Hurkyl]

But that seems paradoxical, since when the cut defined by the same sets is made on the rationals, it defines/creates a real number greater than 0 and less than all numbers of the form 1/m where m = 1,2,3..., according to my (presumably flawed) interpretation of the method for defining the real numbers in terms of Dedekind cuts.

Your interpretation is flawed. It defines a cut equal to zero. In the completed ordering, the point named by (A,B) is the least upper bound for A and the greatest lower bound for B.

Cuts of form #1 always name the maximum of A, and of form #2 name the minimum of B. Only form #3 gives a new point that is strictly greater than every element of A and strictly less than every element of B.

 

[Rasalhague]

Your interpretation is flawed. It defines a cut equal to zero. In the completed ordering, the point named by (A,B) is the least upper bound for A and the greatest lower bound for B.

I don't understand why it defines a cut equal to zero. Not, at least by Goldblatt's definition, surely? What does "complete ordering" mean? Is Goldblatt's definition incompatible with this case? It seems just like Stroyan #3. A and B partition the rationals into two sets. Every element of A is less than every element of B. Why doesn't this define a real number between A and B? Neither author uses the terms least upper bound, greatest lower bound in their definitions. While I'm wresting with those concepts, can you tell me: are both definitions ambiguous because of this, or can my misinterpretation be explained in the same terms the definitions themselves use?

The definitions that I've been looking at just now for least upper bound all define it in terms of real numbers. Presmably there is a way to define real numbers by Dedekind cuts without basing the definition of Dedekind cuts, circularly, on the concept of real numbers.

Cuts of form #1 always name the maximum of A, and of form #2 name the minimum of B. Only form #3 gives a new point that is strictly greater than every element of A and strictly less than every element of B.

Well, Stroyan doesn't explicitly name a maximum or minumum, so I think you're saying this must be a cut of form #3 in the reals, and presumably the new point given, when the cut is made in the reals, would have to be in something else, not the reals. But he says the same sets can be used to make a cut in the rationals. And both Stroyan and Goldblatt define real numbers in terms of the rationals by just such a cut. So it seems this point is a real number when we define it by a Dedekind cut of form #3 on the rationals, but mysteriously stops being a real number when we make the same cut in the reals. Except Stroyan says that it definitely isn't a real number, so obviously I'm still not getting it...

 

[Rasalhague]

I think he just means there is always a fraction smaller than the smallest real.
Take $S = \{1/m: M \in \mathbb{N}\}$, then he defines sets A and B as follows:
- elements of A are smaller than all elements in S
- for every element of B there is an element of S that is smaller

So, there is always an element of S smaller than all elements of B; always a fraction smaller than the smallest real.

But that element of S will be of the form 1/m for some positive integer m. That element of S will be greater than 1/(m+1), and thus itself an element of B. If it was smaller than all elements of B, it would be smaller than itself, a contradiction.

Also, fractions are a subset of the reals. If there was always a fraction smaller than the smallest real, there would be a real smaller than the smallest real. Another contradiction.

 

[gerben]

But that element of S will be of the form 1/m for some positive integer m. That element of S will be greater than 1/(m+1), and thus itself an element of B. If it was smaller than all elements of B, it would be smaller than itself, a contradiction.

It is as simple as this: for every element b ∈ B, S contains per definition an element s that is smaller than b. Lots of elements of S are also elements of B, so what.

Also, fractions are a subset of the reals. If there was always a fraction smaller than the smallest real, there would be a real smaller than the smallest real. Another contradiction.

There is no smallest real. There are reals in between every pair of fractions and fractions in between every pair of reals, there is no contradiction here.

 

[Rasalhague]

It is as simple as this: for every element b ∈ B, S contains per definition an element s that is smaller than b. Lots of elements of S are also elements of B, so what.

I think we can make that stronger than "lots of". Stroyan: "The set B consists of all numbers b such that there is a natural number m with 1/m < b." Your S conists of all numbers of the form 1/m. For all numbers of the form 1/m, there is a natural number, m + 1, such that 1/(1+m) < 1/m. Therefore S is a subset of B. All elements of S are in B.

There is no smallest real. There are reals in between every pair of fractions and fractions in between every pair of reals, there is no contradiction here.

You wrote, "So, there is always an element of S smaller than all elements of B; always a fraction smaller than the smallest real." (I'm guessing from the context you mean "smallest nonzero real".) So this seems to be a third contradiction. But I suppose what your saying is that some ideas that are contradictory in colloquial language, or which contradict untrained human intuition, have to be defined as uncontradictory when we accept any kind of infinity as a topic of discussion? For each element of B, there is an element of S less than it. But can we really say that of all elements of B? If there exists an element s of S smaller than all elements of B, then there exists an element b of B, namely b = s, smaller than all elements of B, including itself. Intuition fails, but this doesn't only contradict intuition, it contradicts Stroyan's ordered field axioms on p. 6, specifically "Every pair of numbers a and b satis es exactly one of the relations a = b, a < b, a > b." In this case, s = b, but you claim that also s < b.

 

[Rasalhague]

In a nutshell: if A is the set of all rationals a satisfyng a < 1/m for all counting numbers m = 1,2,3..., and B is the set of all rationals b such that there exists a counting number m with 1/m < b, why does this not define a real number greater than every element of A and less than every element of B?

"A real number is a Dedekind cut in the set Q of rational numbers: a partition of Q into a pair of nonempty disjoint subsets (A,B) with every element of A less than every element of B and A having no largest member."

Which of the conditions is not met in this example?

(1) The particular A and B in this example partition Q into a pair of nonempty disjoint subsets. (Nonempty is redundant here, isn't it, given that a partition has to cover Q and Q is nonempty?)

(2) Every element of A is less than every element of B. Proof: if there was an a in A greater than or equal to an element b of B, there would be an m such that 1/m < a, but then this a wouldn't meet the condition for being in a. Therefore there is no such a.

(3) A has no largest member. Ah, is this the condition that's not met? Although you (Hurkyl) said "Cuts of form #1 always name the maximum of A, and of form #2 name the minimum of B", could the largest member of A be 0, perhaps named in some implicit sense by the conditions; if so, is it possible to prove this or state it more rigorously that I have?

 

[Rasalhague]

There are three cases for a Dedekind cut in Stroyan's convention:

1. A has a maximum element. B does not have a minimum element.
2. B has a minimum element. A does not have a maximum element.
3. Neither A has a maximum nor B has a minimum.

Cuts of form #1 always name the maximum of A, and of form #2 name the minimum of B. Only form #3 gives a new point that is strictly greater than every element of A and strictly less than every element of B.

Goldblatt's definition states only that A should have no largest member, so cases #2 and #3 qualify as Dedekind cuts for him, but apparently not #1. But for Stroyan, a Dedekind cut "in an ordered field is a pair of nonempty sets A and B so that: every number is in either A or B, every a in A is less than every B in B." This seems to admit the possibility of a type

4. A has a maximum element. B has a maximum element.

(But maybe this type happens not to occur in the rationals, reals and hyperreals due to some property of these particular fields.) And he says, "Dedekind's approach is to let the real numbers be the collection of all cuts in the rational line." And "Dedekind's real numbers fill all such gaps." Is this equivalent to his axom 1.5: "The real numbers are an ordered field such that if A and B form a cut in those numbers, there is a number r such that r is in either A or in B and all other the numbers in A satisfy a < r and in B satisfy r < b"? (A type #3 cut, but this time performed in the reals themselves, rather than a cut of any type, #1, #2, #3 or #4, performed in the rationals.)

It seems that Stroyan's example (the one discussed in my previous post) does define a real number according to Stroyan's own first definition, but not according to Goldblatt's. It looks symmetrical, very much like his definition of 2^1/2 by a cut of the rationals. Is there some important difference between this example and his definition of 2^1/2, an implicit asymmetry between the definitions of A and B in this example that isn't present in the case of 2^1/2?

Could we reword the definitions of A and B in Stroyan's example, changing A to the set of all rational numbers which are less then every element in B; would that qualify as a Dedekind cut according to Goldblatt's definition? (I.e. Stroyan's type #3.) In other words, what kind of expression qualifies as "naming" (in the sense of post 5)?

Another definition of Dedekind cut: "A partition of a sequence into two disjoint subsequences, all the members of one of which are less than all those of the other. The device is used to define the irrational numbers in terms of pairs of sequences of rationals" (Borowski & Borwein: Collins Dictionary of Mathematics). This one, like Stroyan's, seems not to make any requirement about whether one, both or neither of the subsequences has a maximum or minumum.

 

[Hurkyl]

(But maybe this type happens not to occur in the rationals, reals and hyperreals due to some property of these particular fields.)

In a field, every pair of numbers has an average.

And he says, "Dedekind's approach is to let the real numbers be the collection of all cuts in the rational line." And "Dedekind's real numbers fill all such gaps." Is this equivalent to his axom 1.5: "The real numbers are an ordered field such that if A and B form a cut in those numbers, there is a number r such that r is in either A or in B and all other the numbers in A satisfy a < r and in B satisfy r < b"? (A type #3 cut, but this time performed in the reals themselves, rather than a cut of any type, #1, #2, #3 or #4, performed in the rationals.)

I assume when you said "type #3" it's a typo, since that quote said it must be of type #1 or type #2.

The answer is yes or no, depending on what detail you find important. The ordered field Dedekind constructed has the property (called Dedekind completeness) that all cuts are of type #1 or type #2.

For any field of real numbers and field of rational numbers, there is a standard way to view any rational number as a real number. It is true that every real number is either:

• rational, and identified by exactly one rational Dedekind cut of type #1, and exactly one rational Dedekind cut of type #2, or
• irrational, and identified by exactly one rational Dedekind cut of type #3.

In both cases, the real number identified by (A,B) is the unique one that upper bounds A and lower bounds B.

However, generally speaking, it would be wrong to say the real number "is" the Dedekind cut. (or the equivalence class of Dedekind cuts) But this field of real numbers is isomorphic to Dedekind's field of real numbers, and it's easy to switch back and forth between the two.

e.g. given any Euclidean line and choice of two points on it, appropriate +, *, and < operations can be defined so that the set of all points on the line together with those operations is a field of real numbers.

 

[Rasalhague]

I assume when you said "type #3" it's a typo, since that quote said it must be of type #1 or type #2.

Yes, or a brain-O ;-)

The answer is yes or no, depending on what detail you find important. The ordered field Dedekind constructed has the property (called Dedekind completeness) that all cuts are of type #1 or type #2.

For any field of real numbers and field of rational numbers, there is a standard way to view any rational number as a real number. It is true that every real number is either:

• rational, and identified by exactly one rational Dedekind cut of type #1, and exactly one rational Dedekind cut of type #2, or
• irrational, and identified by exactly one rational Dedekind cut of type #3.

In both cases, the real number identified by (A,B) is the unique one that upper bounds A and lower bounds B.

Wow, this is actually starting to makes sense to me! Thanks, Hurkyl.

So in this example, we have a cut of type #1, with maximum element 0, and because A has a maximum element it doesn't define an irrational number. And it could be reworded (perhaps by replacing 1/m with -1/m, and reversing the inequalities) in such a way that it becomes a type #2 cut, with 0 the minimum element of B, and that would also be a representation of 0.

And there is no rational element a of A such that 0 < a because if there was, it would be of the form 1/m which is not less that all rationals of the form 1/m, a contradition. Therefore 0 must be the maximum of A.

And Goldblatt's system just relied on types #2 and #3, which explains why his system doesn't need any further rule to pick out equivalence classes of a cut belonging to type #1 and a cut belonging to type #2.

However, generally speaking, it would be wrong to say the real number "is" the Dedekind cut. (or the equivalence class of Dedekind cuts) But this field of real numbers is isomorphic to Dedekind's field of real numbers, and it's easy to switch back and forth between the two.

Would it be better (than saying a Dedekind cut "is" a real number) to say that the real numbers are an equivalence class of mathematical structures isomorphic to the field defined in this way by Dedekind cuts?

e.g. given any Euclidean line and choice of two points on it, appropriate +, *, and < operations can be defined so that the set of all points on the line together with those operations is a field of real numbers.

Would we also need to identify/label at least two points (so that we know where 0 and 1 are) before letting this Euclidean line based definition into that equivalence class of representations of the real field?

In a field, every pair of numbers has an average.

I don't understand the connection. How about the finite field F₅ with arithmetic modulo 5. Can we not partition this into sets A = {0, 1, 2}, B = {3, 4}, where A has a maximum element, 2, and B a minimum element, 3?

 

[Hurkyl]

Would it be better (than saying a Dedekind cut "is" a real number) to say that the real numbers are an equivalence class of mathematical structures isomorphic to the field defined in this way by Dedekind cuts?

Yes, at least in the back of our minds. This distinction doesn't often have practical relevance, which is why you don't see it explicitly stated very often.

Would we also need to identify/label at least two points (so that we know where 0 and 1 are) before letting this Euclidean line based definition into that equivalence class of representations of the real field?

Ah, the two points I said should be chosen were meant to be 0 and 1 respectively. (Also, for clarity, there are lots of ways to give a "field of real numbers" structure to the line -- I intended to specifically refer to the 'usual' one, however)

I don't understand the connection. How about the finite field F₅ with arithmetic modulo 5. Can we not partition this into sets A = {0, 1, 2}, B = {3, 4}, where A has a maximum element, 2, and B a minimum element, 3?

Any ordered field must have characteristic zero. If F₅ could be ordered as a field, then 0<4 would imply, by adding 1 to both sides, that 1<0.

Actually, the argument I meant still applies too, because if 0<2<3 and 2, then we must have

$$2 = \frac{2 + 2}{2} < \frac{2 + 3}{2} < \frac{3 + 3}{2} = 3$$​

Amongst fields, order is a property enjoyed only by "formally real" fields, ones arithmetically similar to the real numbers. Not only are cannot finite fields and fields of nonzero characteristic be ordered, but the p-Adic fields cannot be ordered either! The complexes can't be ordered either, although its usual topology is closely related to the ordering on the reals.

 

[Rasalhague]

Ah, the two points I said should be chosen were meant to be 0 and 1 respectively. (Also, for clarity, there are lots of ways to give a "field of real numbers" structure to the line -- I intended to specifically refer to the 'usual' one, however)

Oh sorry, of course... I somehow managed to overlook your mention of two points. Thanks again for all your help.