Dedekind's Axiom

Canute

Well, this is just my problem. The axiom states of itself that it applies to a line made of points, or a line of points. Yet a line of points does not seem able to behave in the way the axiom states. If the axiom defines a continuous line by stating that it's a series of points then I will be confused, but assume that the contradiction involved is down to the difficulty of representing the continuous mathematically. Is that it?

That seems ok to me, since you do not state anywhere that there is a unique rational dividing the two classes. What I can't handle is the word 'all' at the start of the axiom. If all means all then where does the unique dividing point come from?

In oversimplified terms is the axiom saying that between any two points there is another point? That's my naive reading of it. Or is it not a stand alone axiom, and so perhaps my problem stems from not knowing the others that go with it?

But maybe I need to understand what an axiom of completeness is. I know this is painfully elementary but could you briefly explain? Sorry to be a bore but I want to understand this. After what's been said I'm not even sure now whether the axiom defines a series of points or a continuum, even though I started out taking it at face value, as the former.

matt grime

an axiom is jsut an axiom, for instance the axiom

G contains a unique element e such that e*x=x*e=x for all x

is an axiom from group theory.

it itself has no truth value and is only true of false when applied to some G and *.

here dedekind's axiom is not true in and of itself or true for all "lines". it only makes sense to say it is true or false when applied to something. and it is true for the real line and false fro the rationals. the reals are complete (all cauchy sequences have a limit) and the rationals aren't, ie it is possible to find a sequence of rationals increasing and bounded above that do not converge to a rational.


the axiom states that if we define two classes of elements of the reals, one allto the left of the other that this is the same as picking a unique point of division.

thus the example i gave abuot the positive rationals whose square is bigger than 2 and its complement divide the line into two parts, but there is no rational that is the point of divisoin.

another way is this: if we define two classes dividing the line up then we an take the sup of the left hand one and the inf of the right hand one and these agree, and that this element is in either the left or right hand sets.

bao_ho

all is all, and if is if.

Canute

So is the dividing point both the sup of the LH class and the inf of the RH class, or is it just one of them? If so which one? But there's no need to answer that, the axiom still makes no sense to me but I've probably pushed your patience far enough. At the moment I rather agree with bao ho, but I'll go away and do some reading around it. Sorry to be so dense but thanks for your help.

AKG

It's both the sup of the LH class and the inf of the RH class. I can't understand how this axiom is troubling you. For one, did you even see what matt wrote about axioms? They're axioms. It's not a statement of a law of nature. If I define a word "spen" as a pen that has a mismatched cap, will you say, "that definition makes no sense to me?" Dedekind's axiom helps to define the reals, so the reals are by definition, some set of numbers that is, put briefly, complete. So I hope you don't disagree that the Reals are complete, since that wouldn't make sense. Do you just not understand what it means to be complete? What is it that's so confusing?

Suppose you have the integers. If you cut the integer line at the point 2.5, then you'll split the integers into two classes, the class A = {..., -3, -2, -1, 0, 1, 2} and the class B = {3, 4, 5, ...}. Here, every point of class A is to the left of class B. However, if you divided the line at 2.75, you'd have the same two classes, but with a different dividing point. So the integers are not complete. In the case of the reals, is it evident to you that any splitting of the Reals into two classes such that all of one class is less than all of the other must split in one of the following two ways:

(-infinity, x) and [x, infinity)

or

(-infinity, x] and (x, infinity)

Is it also not obvious to you that in these cases, there is one and only one dividing point, and that point is x?

Zurtex

They're both the same uniquie dividing point!

bao_ho's post makes no sense other than refering to the definition of the words!

matt grime

one thing about AKG's post is that to "split the integers to the left and right of 2.75" assumes thart such as thing as 2.75 exists, but it may not, well it does, but we dont' know that, or more accurately the completeness axiom doesn't say what does or deosn't exist. It may be that *nothing* satisfies the completeness axiom. As we can show there is something that does satisfy the axiom.


ok to be honest i *wanted* to use this kind of example but thought of the several ontological issues. actually the integers are a bad example to use since in any reasonable sense they are complete since a cauchy sequence is eventually constant and hence converges. but the integers aren't a field.

let's assume that we konw the reals exist and the rationals are inside them


define a splitting of Q via L as the rationals less than pi and R as the rationals greater than pi.

sup(L) and inf(L) are not in Q, there is no rational that is the cut point. the rationals are not complete.

what is hard about this?

AKG

I thought of this too, but I don't see how it's a bad example. As Canute is having trouble grasping this simple axiom, I figured that this example would be more instructive even if it isn't technically that good. Is he going to know what a Cauchy sequence, or even a field is?

Also, if it's bad to split the integers by 2.75, then it's bad to split the rationals by pi. I think if you wanted, you could introduce surreal numbers and split the reals into classes where the splitting point is not in the reals.

matt grime

The difference between the rationals and the integers is that the integers are complete. You need at least the rationals for this to make sense.

I can split the rationals by sqrt(2) without mentioning the 'ambient' space of reals. as i have done several times by example. Really we must only define our split by reference to the underlying space, so i can state things like the integers n such that 2n<5 but even then the sup exists over the integers and is an element of the left hadn set.

The rational example I gave was the positive rationals x suich that x^2>2 and its complement in the rationals. there is no rational splitting point.

if you don't know about cuachy sequences or completeness what is the point of dedekind's axiom? (this isn't rhetorical: i never learned abuot the construction of the reals until i had to teach it. if you like it was something we never mentioned as a student and by the time i realized (teaching at a different place) that it wasn't obvious it was too late....)

Canute

I'm aware of what an axiom is. My query is not whether this axiom applies to reality or not, that doesn't matter. But it appears to me, even now, that it is internally inconsistent. In the example dividing the integers, 2.5 was said to be the dividing point, but 2.5 is not a member of the two classes being divided so this does not seem to be an equivalent case.

What you all seem to be saying is that the axiom does not stand alone but assumes certain other prior axioms, so is my problem that I'm reading it in isolation from these? I'm reading it naively, as a stand-alone statement about a line of points and the nature of the number line.

Zurtex - "They're both the same unique dividing point!"
This is what I thought was being said, but to me it seems an oxymoronic statement.

AKG

I still don't get what's troubling you. Where is the internal inconsistency? To me, the axiom does seem to be lacking a little, but I don't see it as inconsistent. The axioms says that if you split a line into a left half and a right half, that there exists a unique dividing point. What confuses you? That there exists a dividing point, or that this point is unique?

selfAdjoint

The axiom says the two sets exhaust the reals, i.e. every real number is in one set or the other. And every number in B is greater than every number in A. So the cut will be a member of B, less than every OTHER member of B, but sharing the B property of being greater than every number of A.

Consider A the set of numbers whose squares are less than 2. And let B be the complement of A, consisting of all other numbers. Then the axiom says there is a LEAST member of B, call it c, characterized by a < c < b, for any member a in A and other b in B. This number c then has square = 2. For it is not in A so its square is not less than 2, and if its square were greater than 2, there would be some other number whose square was equal to 2, and then that number would be in B, not A, and would be less than c. But c is the least member of B, so that's a contradiction.

AKG

selfAdjoint, the cut point could be a member of A. Also, the set A of numbers whose square is less than 2 is (-sqrt(2), sqrt(2)). This divides the reals into 3.

matt grime

you say this but you appear not to understand it

and the axiom is false for the rationals. this example was supposed to show you why it applies to the real numbers and not other sets of numbers that happen to form a line.

no, it may stand alone as may any axiom, but it only becomes meanignful to see if for some mathematical objesct the axiom is true for it. here is anothe example (axiom of algebraic completeness) a field k is algebraicaly closed if every polynomial p(x) with coefficients in k has a root in k. This axiom is true for the complex numbers but false for the real numbers.

read it however you will but the important point is that the reals (being complete) are such that the axiom is true for them, The rationals it is not true for. is that clear?

Canute

You say that this axiom applies to this sort of number and not that sort of number and so on, but the axiom does not mention numbers at all, it mentions only points. You've told me that the axiom does not require any supporting axioms. In this case why are we talking about numbers and not points? Points cannot be distinguished from one another as if they were reals and rationals.

Let's say we wanted to represent time mathematically, as a series of moments or points in time. If we have to apply Dedekind's axiom in order to do this then we have to say that all moments fall into one of two classes, with every moment in the first class lying in the past relative to any point in the second class. But in which class does the present moment belong? It hardly matters whether we call these points in time real or rational numbers, we could represent them by the letters of the alphabet if we wanted.

This is the issue I'm delving into. This is why I said I wasn't quite sure the question belongs here, thinking it might belong in philosophy of mathematics. I'm happy to move it there if you think that's a better place for it.

matt grime

becuase the whole raison' d'etre of dedekind's axiom, along with dedekind cuts, is to define properly the real numbers. i thought you were attempting to understand what or where it came from and "what it applies to" and as such were aware of its implications in the construction of the real numbers. we are trying to illustrate its implications by example, indeed the only example that is important. creating the dedekind cuts from the set or rationals creates a (real) totaly ordered field that satisfies the dedekind axiom.

As for the rest, time whatever it ought to be (continuous, discrete, whatever) has no point in this discussion. all you are asking is "is time better modelled by the real numbers or something else?" and is not a question that has anything to do with mathematics.

AKG

Looking back, you seem to think the contradiction is that Dedekind's axiom says there are two classes, but then introduces a point which you believe is of neither class. But his axiom does not say that it is of neither class. The axiom, as you've stated it, is actually a little ambiguous. It refers to it as the point that "produces the division."

Anyways, if you are going to divide time then in your example, you could either divide it into one class of points that are strictly before the present, and one class that is before or during the present OR divide it into one class of points that are before and including the present, and one that is strictly after the present. Do you see how these would be complete divisions? If we took two classes, one that was strictly before and one that was strictly after, we wouldn't have completely divided time. Now, regardless of which of the above two ways you divide time, the present serves as the one and only dividing point. And depending on how you do this, the dividing point is either in the earlier class or the later class, but it is going to be in one of them, it's not in a third class. So what's the problem?