Dedekind Cuts

I read about how the union of dedekind cuts can form the reals...but I don't really understand why...

A dedekind cut has the form {r in Q: r < a } for a in R...

Since r needs to be in the rationals, it cannot be number such as the square root of 2, right? Then how can they form the reals?

Can anybody clarify this to me?

Thanks in advance.

Reply 1

DFranklin

18

A Dedekind cut is a set, not a single number. For example, we can write sqrt(2) as the set

S = \{x \in \mathbb{Q} : x < 0 \text{ or } x^2 < 2 \}

. Then every element of S is rational, but sup(S) = sqrt(2).

Reply 2

DFranklin

18

Artus

..quotiness..

By the way, I would consider every question you've posted recently as "undergraduate" level rather than postgrad, so I would label them as such if I were you.

Reply 3

atsruser

11

Original post by Artus

I read about how the union of dedekind cuts can form the reals...but I don't really understand why...

A dedekind cut has the form {r in Q: r < a } for a in R...

Since r needs to be in the rationals, it cannot be number such as the square root of 2, right? Then how can they form the reals?

Can anybody clarify this to me?

Thanks in advance.

The idea of a Dedekind cut is to construct the real numbers from the rational numbers. It uses set theory. The idea, roughly, is that we define a set

D

of rational numbers which has no maximum value, but whose rational elements can get arbitrarily close to that maximum (similar to the idea of an open set).

In addition, if the set

D

contains

y \in \Bbb Q

and

x < y

then

x \in D

. (So this tries to formally say that

D

contains all of the rational numbers less than the maximum, going off to

-\infty

).

So for example the set:

D = \{x \in \Bbb Q: x < 0 \cup x^2 < 2\}

is a Dedekind cut. You can show that if you have

x \in D

and

x^2 < 2

then you can always find

y \in D

and

y > x

and

y^2 < 2

i.e. there is no maximal element of

D

.

A cut corresponds to our idea that each real number

r

splits the number line into two parts, with one containing all the numbers less than

r

and the other containing all the numbers greater than it.

Dedekind then used these cuts to define the real numbers. He said, for example, that

D

defines

\sqrt{2}

.

To do this completely, you need to show that there are set operations that you can perform on the cuts (which are of course sets), which corresponding to the normal operations of addition, subtraction, and multiplication.

So for example, if cuts

X

and

Y

represent real numbers x and y then

X \subset Y

corresponds to the fact about the real numbers: x < y

Also, we can define addition of real numbers, by addition of the rationals in each cut i.e. the number x + y corresponds to the set

X+Y= \{a +b : a \in X \cap a \in Y \}

. (This is itself a cut)

You can also define multiplication in terms of cuts, though I forget the details (Google is your friend).

Finally, and perhaps the most important point: if we consider the real numbers as the union of *all* Dedekind cuts in the rationals, then that set is complete: i.e. all sets which are bounded above have a least upper bound (a supremum) So for example there is no least upper bound for the elements of set

D

in the rationals, though it's certainly bounded above (by 20, say), but the cut itself defines the least upper bound, and we consider it to be the real number

\sqrt{2}

.

Does that help?

Reply 4

Artus

OP

Original post by DFranklin

A Dedekind cut is a set, not a single number. For example, we can write sqrt(2) as the set

S = \{x \in \mathbb{Q} : x < 0 \text{ or } x^2 < 2 \}

. Then every element of S is rational, but sup(S) = sqrt(2).

Sup (S) = sqrt(2), but the supremum is not contained in the set, right? I still don't understand how the union of dedekind cuts gives us the real numbers. Since all dedekind cuts contain only rationals, their union must also just contain rationals...right? :confused:

Reply 5

nuodai

17

Original post by Artus

Sup (S) = sqrt(2), but the supremum is not contained in the set, right? I still don't understand how the union of dedekind cuts gives us the real numbers. Since all dedekind cuts contain only rationals, their union must also just contain rationals...right? :confused:

Don't confuse a set of sets with the union of the sets in a set of sets.

Let me clarify. A Dedekind cut is a set

D \subseteq \mathbb{Q}

satisfying a certain property. We can then define

\mathbb{R} = \{ D \subseteq \mathbb{Q}\, :\, D\ \text{is a Dedekind cut} \}

whereas what you have interpreted this to mean is

\mathbb{R} = \displaystyle \bigcup \{ D \subseteq \mathbb{Q}\, :\, D\ \text{is a Dedekind cut} \}

(which is actually the union of all the members of

\mathbb{R}

, where we consider

\mathbb{R}

to be a set of subsets of

\mathbb{Q}

as given by the first definition I gave).

That is, when constructed in this way,

\mathbb{R}

is a set of subsets of

\mathbb{Q}

- its elements are not the elements of these subsets; instead, its elements are the subsets themselves.

Note that this definition is entirely formal (of course - we want to think of the elements of

\mathbb{R}

as being 'numbers' but here we're saying that real numbers are 'subsets of the rationals' - wtf?). In particular, we'd like to think that

\mathbb{Q} \subseteq \mathbb{R}

but this definition does not allow for that (but it does allow for an inclusion

\mathbb{Q} \hookrightarrow \mathbb{R}

which maps a rational number

q

to the Dedekind cut whose supremum is

q

). So to get back from this weird 'set of sets' definition to what we think real numbers are, we can just identify what we think a real number is with the Dedekind cut whose supremum is what we think this real number is.

Sorry if I've made no sense. There are few ways of explaining this kind of thing without convoluted discourse.

(edited 11 years ago)

Reply 6

atsruser

11

Original post by Artus

Sup (S) = sqrt(2), but the supremum is not contained in the set, right? I still don't understand how the union of dedekind cuts gives us the real numbers. Since all dedekind cuts contain only rationals, their union must also just contain rationals...right? :confused:

Dedekind was trying to come up with a set of objects which behave in a similar way to the real numbers, but starting from the position that

\mathbb{Q}

was well-defined and understood i.e. that mathematically it forms an ordered field, a set where you can tell which of two rationals are larger, and where you can add/subtract and multiply/divide them.

Dedekind was well aware that there are non-rational real numbers like

\sqrt{2}

. He was well aware that

\mathbb{Q}

was incomplete i.e. that there were some subsets of

\mathbb{Q}

which were bounded above, but which did not have a least upper bound. For example, the set I gave in my last post is such a one:

D = \{x \in \Bbb Q: x < 0 \cup x^2 < 2\}

The elements of this set can get arbitrarily close to the number that we think of as

\sqrt{2}

, but sadly that number is not rational, so there is no least upper bound for the set

D

that can be found in

\mathbb{Q}

.

Dedekind therefore had to invent some set of objects which:

1. were defined in terms of the rational numbers and nothing else.

2. behaved liked the rational numbers (i.e. could be considered as an ordered field).

3. was a complete set i.e. contained the "missing" irrational numbers that allowed all sets that were bounded above to have a least upper bound (a supremum).

The final point is the most important, as the idea of completeness is that which corresponds to our intuitive understanding of

\mathbb{R}

as having no gaps; mathematicians have decided that this is well modelled by requiring that sets which are bounded above to have least upper bounds. If we don't have complete sets, we can't define continuous functions, for example.

He had the brilliant idea to invent the concept of a "cut", and to define it formally in terms of members of

\mathbb{Q}

. He now had to define a set of operations on the cuts (these must be set operations, of course, like inclusion, union and so on, since a cut is a set) which could be mapped onto the usual numeric operations that we perform with the rationals.

1. He showed that set inclusion of one cut in another can be mapped to the ordering operation

<

2. He showed that there were set operations that could be defined that mapped onto addition and multiplication

3. He showed that the set of all cuts was complete i.e. that for any non-empty subset of the set of all cuts, you could find a particular cut that behaved as the supremum of that non-empty subset.

4. Because of point 3, in particular, Dedekind said that we could identify the set of all cuts as being equivalent to the set of real numbers

\mathbb{R}

.

It's important to be clear in your mind as to what's going on: we're not saying that that the set of all cuts "is" the real numbers; rather we're saying that for any operation that you reasonably want to perform on the real numbers, there is a corresponding operation that you can perform on the set of cuts that behaves in a similar way to that operation on the reals.

We come across a similar idea in group theory, that of isomorphism, where we have two different sets and two different group operations, but we can formally define a bijection between the two sets which maps the group operations in the "same" way (look up isomorphism if you're not sure what I mean here).

In the case of the "mapping" between cuts and real numbers, we have a similar but more complex situation, since we have more operations to map: ordering, addition, multiplication, and completeness. However, we have on the one side a little world of sets (the cuts) which we can combine in various ways, and on the other, a little world of real numbers which we can combine in various ways, and Dedekind pointed out a complex set of mappings from one world to the other which allows to recognise that the two worlds, cuts and real numbers, behave in exactly the same ways. That is why we say that the cuts define the real numbers: really we mean that they model the real numbers exactly.

(One final point: presumably the "mapping" defined by Dedekind has a formal name in some branch of mathematics, but I don't know what it is).

(edited 11 years ago)

Reply 7

Artus

OP

Original post by atsruser

Dedekind was trying to come up with a set of objects which behave in a similar way to the real numbers, but starting from the position that

\mathbb{Q}

was well-defined and understood i.e. that mathematically it forms an ordered field, a set where you can tell which of two rationals are larger, and where you can add/subtract and multiply/divide them.

Dedekind was well aware that there are non-rational real numbers like

\sqrt{2}

. He was well aware that

\mathbb{Q}

was incomplete i.e. that there were some subsets of

\mathbb{Q}

which were bounded above, but which did not have an upper bound. For example, the set I gave in my last post is such a one:

D = \{x \in \Bbb Q: x < 0 \cup x^2 < 2\}

The elements of this set can get arbitrarily close to the number that we think of as

\sqrt{2}

, but sadly that number is not rational, so there is no least upper bound for the set

D

that can be found in

\mathbb{Q}

.

Dedekind therefore had to invent some set of objects which:

1. were defined in terms of the rational numbers and nothing else.

2. behaved liked the rational numbers (i.e. could be considered as an ordered field).

3. was a complete set i.e. contained the "missing" irrational numbers that allowed all sets that were bounded above to have a least upper bound (a supremum).

The final point is the most important, as the idea of completeness is that which corresponds to our intuitive understanding of

\mathbb{R}

as having no gaps; mathematicians have decided that this is well modelled by requiring that sets which are bounded above to have least upper bounds. If we don't have complete sets, we can't define continuous functions, for example.

He had the brilliant idea to invent the concept of a "cut", and to define it formally in terms of members of

\mathbb{Q}

. He now had to define a set of operations on the cuts (these must be set operations, of course, like inclusion, union and so on, since a cut is a set) which could be mapped onto the usual numeric operations that we perform with the rationals.

1. He showed that set inclusion of one cut in another can be mapped to the ordering operation

<

2. He showed that there were set operations that could be defined that mapped onto addition and multiplication

3. He showed that the set of all cuts was complete i.e. that for any non-empty subset of the set of all cuts, you could find a particular cut that behaved as the supremum of that non-empty subset.

4. Because of point 3, in particular, Dedekind said that we could identify the set of all cuts as being equivalent to the set of real numbers

\mathbb{R}

.

It's important to be clear in your mind as to what's going on: we're not saying that that the set of all cuts "is" the real numbers; rather we're saying that for any operation that you reasonably want to perform on the real numbers, there is a corresponding operation that you can perform on the set of cuts that behaves in a similar way to that operation on the reals.

We come across a similar idea in group theory, that of isomorphism, where we have two different sets and two different group operations, but we can formally define a bijection between the two sets which maps the group operations in the "same" way (look up isomorphism if you're not sure what I mean here).

In the case of the "mapping" between cuts and real numbers, we have a similar but more complex situation, since we have more operations to map: ordering, addition, multiplication, and completeness. However, we have on the one side a little world of sets (the cuts) which we can combine in various ways, and on the other, a little world of real numbers which we can combine in various ways, and Dedekind pointed out a complex set of mappings from one world to the other which allows to recognise that the two worlds, cuts and real numbers, behave in exactly the same ways. That is why we say that the cuts define the real numbers: really we mean that they model the real numbers exactly.

(One final point: presumably the "mapping" defined by Dedekind has a formal name in some branch of mathematics, but I don't know what it is).

Thanks a lot. I think I understood it better now. :smile:

Reply 8

BabyMaths

Original post by atsruser

... which were bounded above, but which did not have an upper bound...

Sorry to pick on a tiny bit of a long post.

Reply 9

atsruser

11

Original post by BabyMaths

Sorry to pick on a tiny bit of a long post.

Thank you. Now corrected.

Reply 10

john_gabriel

Most of the previous answers are either wrong or partly wrong.

In fact, D. cuts do not define real numbers contrary to popular opinion.

To see how easily one can debunk D. cuts, visit the following page and read comment #3163:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html

In this comment, I define D. cuts for irrational numbers using constructive mathematics and then proceed to show that the idea is indeed absurd, that is, it fails to define the imaginary objects one thinks of as "irrational numbers".

Reply 11

FireGarden

14

Original post by john_gabriel

Most of the previous answers are either wrong or partly wrong.

In fact, D. cuts do not define real numbers contrary to popular opinion.

To see how easily one can debunk D. cuts, visit the following page and read comment #3163:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html

In this comment, I define D. cuts for irrational numbers using constructive mathematics and then proceed to show that the idea is indeed absurd, that is, it fails to define the imaginary objects one thinks of as "irrational numbers".

You dug up a year old post to claim dedekind cuts don't define real numbers?

Reply 12

FJacob

Original post by FireGarden

You dug up a year old post to claim dedekind cuts don't define real numbers?

It gets better... in the post he linked, he claims pi is not a number...

By the way, if I can use this revival to ask a question on this topic:
what about transcendental numbers (i.e. non algebraic) that cannot really be used to write down the description of a set? Do you just assume they are included or is there a way to describe every single irrational number with set theory language?

(edited 10 years ago)

Reply 13

atsruser

11

Original post by FJacob

It gets better... in the post he linked, he claims pi is not a number...

I'm glad that you were brave enough to look. It saves me having to do so.

By the way, if I can use this revival to ask a question on this topic:
what about transcendental numbers (i.e. non algebraic) that cannot really be used to write down the description of a set?

I've no idea what you mean by "cannot really be used to write down the description of a set". Can you expand on this?

Do you just assume they are included

Assume that they are included in what? If you mean included in the set of numbers defined by Dedekind cuts, then the answer is yes, since all (real) transcendental numbers are irrational, and all irrational numbers have a corresponding cut.

or is there a way to describe every single irrational number with set theory language?

Again, I don't understand the question.

Reply 14

FireGarden

14

Not sure on transcendentals, however Cauchy sequences (another method of defining real numbers) would be able to handle that case. Take e or instance; defined by a power series, it is then "real" since it can be approximated with arbitrary precision via a sequence based on the partial sums of its power series.

Reply 15

FJacob

Original post by atsruser

I'm glad that you were brave enough to look. It saves me having to do so.

I've no idea what you mean by "cannot really be used to write down the description of a set". Can you expand on this?

Assume that they are included in what? If you mean included in the set of numbers defined by Dedekind cuts, then the answer is yes, since all (real) transcendental numbers are irrational, and all irrational numbers have a corresponding cut.

Again, I don't understand the question.

I'm sorry about that. Reading it again I was certainly not very clear.

When defining a dedekind cut for sqrt(2), you can use x^2<2, and the same thing for other algebraic numbers.
Now, what about a number such as 0.112123123412345...?
I could see, as FireGarden described, it being defined by a sequence, but in terms of dedekind cuts, would there be a way to define this number just using set theory language?
The thing that bothered me is that it is said that dedekind cuts define all reals, but you can only define a dedekind cut with set theory language (i.e. a set that has some properties).
I'm not sure how you would then define cuts for transcendental numbers without using cauchy sequences.

Reply 16

atsruser

11

Original post by FJacob

When defining a dedekind cut for sqrt(2), you can use x^2<2, and the same thing for other algebraic numbers.

Strictly, that doesn't define a cut, since it doesn't include the negative numbers greater than sqrt(2).

Now, what about a number such as 0.112123123412345...?

I guess that number is known to be transcendental?

I could see, as FireGarden described, it being defined by a sequence, but in terms of dedekind cuts, would there be a way to define this number just using set theory language?
The thing that bothered me is that it is said that dedekind cuts define all reals, but you can only define a dedekind cut with set theory language (i.e. a set that has some properties).
I'm not sure how you would then define cuts for transcendental numbers without using cauchy sequences.

I understand what you mean now. You are wondering how we could specify the necessary condition for the cut, similar to what we did for the cut defining sqrt(2), for this more complex case, and if we can't, does that mean that the Dedekind cut for it doesn't exist?

I'm not entirely sure. However, presumably this number (let's call it C) is well-defined, so in principle, for any given rational r, we can tell if r > C or r < C. This procedure may be difficult, but unless it is possible in principle, we can't say that have a good definition for C.

Then we can clearly define the cut for C via

\{r \in \mathbb{Q} : r < C \}

For a better answer, you may want to post the question on math.stackexchange.

Reply 17

john_gabriel

Original post by FJacob

It gets better... in the post he linked, he claims pi is not a number...

By the way, if I can use this revival to ask a question on this topic:
what about transcendental numbers (i.e. non algebraic) that cannot really be used to write down the description of a set? Do you just assume they are included or is there a way to describe every single irrational number with set theory language?

That's correct. I do claim pi is not a number. And you've not proved anything to the contrary. In fact, I doubt you know what it means "to be a number".

Dedekind Cuts

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