Is Dedekind's Axiom Still Relevant in Modern Mathematics?

[matt grime]

I'm aware of what an axiom is.

you say this but you appear not to understand it

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.

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.

What you all seem to be saying is that the axiom does not stand alone but assumes certain other prior axioms,

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.

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.

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]

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.

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 modeled 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?

 

[Canute]

Thanks to everyone for the discussion. I'll leave it there it that's ok because I think we've reached an impasse. I came across the axiom in Dantzig's book on mathematics, much praised by Einstein. He takes the view that the axiom is problematic, and I'm afraid despite what's been said here I still agree with him. But of course it's possible that I've misinterpreted him in some way. Does anyone have a copy? I borrowed it and now can't refer back to it. He specifically discusses the problem of Dedekind's cut in relation to the mathematical treatment of time, and what he says seems right to me. This is why I was suprised at the reaction to my question here. I was just checking my understanding of him and not expecting a disagreement. Probably my fault as a non-mathematician for using the wrong terminology or something.

Regards
Canute

 

[matt grime]

why are you surprised at the response you got here? we are simply treating the axiom in its own right and not thinking of time. why wuold we think of time? we are attemptinf to make you see that the axiom makes perfect sense and is true for some objects, false for others. if you want to discuss the relevance of the real numbers as a suitalbe description of time, and in particular the fact that the reals satisfy dedekinds axiom and when we discuss time as the use of the reals we get an apparent contradiction, fine. there are many such problems with things like the real numbers or R^3 not least of which is the Banach-Tarski paradox. but that wasn't what you were saying (though it may have been what you meant to say) to our eyes.

there may well be problems if we apply the notion of dekekind's axiom to soemthing physical like time. this doesn't make the axiom problematic in and of itself and is "merely" a question about it's applicability to thinking about it in respect of time. ie he is attempting to describe some physical (or possibly philosophical) problem with the use of the real numbers to describe time. exactly the same could be said for attempting to descrbe lengths using real numbers. see the banach tarski paradox for a well known example. the real numbers are a mathematical invention with some very strange properties, stranger than yoe can ever think of (to paraphrase someone whose name i can't remember). so? that doesn't mean there is a *mathematical* issue with the axiom. the real numbers still satisfy dedekind's axiom, the rationals do not. there are many problems with the real numbers and their suitability to be used in the real world but they are principally of a philosphical nature. indeed there is a reasonable claim that the only numbers we ever need are (a subset of) the constructible ones, and they are countable. so how can something like the real numbers, almost all of which are things we can never know, be suitable for describing something physically meaningful like time? we could just as well be talking abuot the square root of 2 and lengths of hypoteneuses.

an axiom many be problematic (a system of axioms cabn be inconsistent in that they may be mutually contradictory) in that it may be vacuous, always false or just plain silly, and it can be true or false when thought of in relation to some model. the set of dedekind cuts of rationals is a model where the axiom is true. this may make some philosophical problem occur, it may not. i don't think anyone here would care to speculate on what the thoughts of some author we may never have heard of meant in an sentence in some book about time and the relevance of the reals to describe them.

 

[Zurtex]

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.

Just to add, this is only oxymoronic if you consider saying that (1 - 1) and (-1 + 1) are both the same number.

Also, just to clarify, you do know the difference between an axiom and a theorem right?

 

[Canute]

Yes, I know the difference the between an axiom and a theorem, as I said before. An axiom is an assumption or postulate and a theorem is a formal derivation from same.

I feel AKG's response comes close to answering my query when he says "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."

That makes sense to me, and it leaves the axiom ambiguous rather than contradictory. (It is or is not ambiguous/paradoxical regardlessw of whether it is an axiom or a theorem). If the dividing point is in one of the classes then that would be a solution to my problem. However, someone stated that the dividing point is unique and that threw me off the track, since this is untrue if the point is in one the classes. If the dividing point is in one of the classes then it cannot be the dividing point between the classes, and if it is one of the classes it is no different to its complement in the other class (sup x = inf x).

I raised the issue of time because it highlighted what I see as the underlying issue, the difficulty of treating a continuous line as a series of points. However, I've read Dedekind's axiom in isolation and taken it at face value. What you are saying is that I cannot do that, that I have to consider it as a statement about the reals or the rationals or whatever, not as a statement about a series of points. If this is so then I've misinterpreted the axiom, since I took it to be a statement about a series of points (an understandable mistake I feel given that this is what it says it is).

I'll assume it's a statement about numbers then, and that it is a formal or heuristic device, to be interpreted in a very particular way as a statement about certain kinds of numbers but not others. In this case I have no problem with it, other than that I wish the axiom itself made it clear that it does not apply to a series of points.

But I'm still slightly confused. Could someone briefly outline the function of the axiom within mathematics, how it is used and why? Or is this too big a topic to deal with here? This would help me understand what it actually states. Thanks for all the responses so far.

 

[matt grime]

let's take the standard example we have been using again.

consider the reals, take them and split them into two parts L and R one to the left of the other as it says in the axiom. In the reals, which satisfy the axiom, we are saying that L and R are one of L=(-inf, x) R=[x,inf) or L=(-inf,x] (x,inf) and x is the unique point creating the division (note there are two ways to split it up). In the rationals the axiom does not hold example: R is the set of positive rationals whose square is bigger than 2, and L its complement. If there were a rational creating this division then it would have to be square root of 2 which we know is not rational, ie there is no rational that is smallest and satisfies the property x^2>2

We can think of the axiom possibly applying to any linearly ordered set (the line of points in the axiom) whence it may be satisfied or not, let;s say that the class of linearly ordered sets that satisfy dedekinds axiom are called dedekind complete. so it may apply to any set of linearly ordered points. just as the axiom 'a binary operation * has a two sided identity' may or may not be true of any given operation.

in this case the reals are 'dedekind complete' (as are the integers) but the rationals are not dedekind complete.

The axiom is equivalent to: given any splitting L,R of the set of points S as in the axiom then there is a unique x in S which is sup(L) and inf(R) and necessarily lies in one of the L and R (and exactly one) since S is the disjoint union of L and R.

you may apply it to the set of points corresponding to time and it appears your author thinks that they do not satisfy the axiom in his view, or the assumption that they do is problematic, ie the set of points of time are not dedekind complete. well, that's fine, I suppose, but it isn' a problem with the axiom really any more than it is a 'problem' that the rationals don't satisfy the axiom. this is a 'problem' with your (his) view of time.

 

[Canute]

Ah, I think I get it now. I'd assumed that because the axiom is an axiom or a theorem in all/most mathematical systems, as was said at the start, that it was taken as true in all these systems as a statement about any line of points. But I was wrong, some classes of linearly ordered sets are not Dedekind complete. So it is an axiom with limited application, a property of some lines of points but not others. Now I'm getting there.

I still have some trouble with idea that x is a point of division between two classes but is nevertheless a member of one class or the other depending on how one splits them. Such a statement wouldn't make much sense in ordinary language, or outside of a formal mathematical scheme. But if it is a useful axiom in some circumstances then fine. I think you've cleared this up for me.

You are probably right to say that Dantzig was arguing that time does not satisfy the axiom. (If I had the book here I'd quote the passage). He seemed to be saying that any mathematics of time must be one in which this axiom does not appear. I had assumed that the axiom was more fundamental to mathematics than it is, so wondered what this implied for mathematics as a whole, as a means of describing reality. If the axiom may or may not apply then it implies nothing much. Thanks for clearing that up, and sorry to take so long to get the message.

 

[Zurtex]

Canute, as I am sure it has been mentioned in this post, it is not the job of mathematicians to describe reality. That's the job of physicists and philosophers, the former of which tend to use maths to do it.

Can time be accurately represented as a real number line or at least a sub set of the real number line with the same cardinality? I’m not convinced physicists will ever really be able to answer that one.

 

[Canute]

I take your point about mathematics and reality. The problem is that physicists use mathematics to describe reality even if mathematicians don't. Still, I understand some of the issues better now having chatted here.

 

[HallsofIvy]

It seems to me that if x is in the first class then the division between the classes occurs to the right of x, and if x is in the second class then the division is to the left of x. In neither case does x represent the point of division between the classes.

No, no, no! The two sets A= (-infinity, 0), B= [0, +infinity) have the property that every member of A is to the left of every member of B and every member of B is to the right of every member of A. The unique number that splits the two sets is 0. You say "if x is in the second class then the division is to the left of x". Where "to the left of x"? The only numbers to the left of 0 are negative numbers and every negative number, y, has another negative number, y/2, to the right of it and so doesn't produce this splitting.

 

[AKG]

That makes sense to me, and it leaves the axiom ambiguous rather than contradictory. (It is or is not ambiguous/paradoxical regardlessw of whether it is an axiom or a theorem). If the dividing point is in one of the classes then that would be a solution to my problem. However, someone stated that the dividing point is unique and that threw me off the track, since this is untrue if the point is in one the classes. If the dividing point is in one of the classes then it cannot be the dividing point between the classes, and if it is one of the classes it is no different to its complement in the other class (sup x = inf x).

Huh? Why can't the dividing point be unique if it's in one of the classes. It's the point that "produces the division" it is not the point that is between both classes. Like I said (I think), "produces the division" is not a technical term, but it's meaning is obvious. The other thing to note that while sup((-infinity, 0)) = 0 = inf([0, infinity)), 0 is an element of [0, infinity) while not an element of (-infinity, 0). That is, the supremum, if it exists, of an open interval of the reals is not in the interval. This is how the supremum of one class is the infimum of the other, without this unique point being in both classes.

 

[Hurkyl]

Here's an example of why one might want to use the completeness axiom for the purposes of physics:


Suppose you have a weight attached to a frictionless spring, sitting at equilibrium. You tug on the weight and then let go, watching it oscillate.

You'd probably want to say something like the motion of the spring is periodic, and you'd probably like to do things like compute its period, or its amplitude.

I challenge you to try and do that without invoking some form of the completeness axiom!


Here's a more formal example of the point I'm trying to make: the function sin x is NOT a periodic function over the rationals: there is no nonzero (rational) number a such that sin x = sin (x+a) is an identity.


The completeness axiom let's us say other "obvious" things too. For example, if we were racing, and if I wanted to say "Well, I was ahead of you at first, but then later you were ahead of me, so there was some point in time at which we were tied...", then (in general) I must invoke some form of completeness axiom to make that conclusion. (And an assumption about the continuity of our trajectories)


Most (all?) of what you learned in calculus simply does not work "properly" if you don't have a guarantee of completeness.

 

[Canute]

This seems to say no more than that the axiom is required to describe motion. But it's just been said that it is not the job of mathematics to describe reality. This latter view seems more correct to me, since while describing motion in this way, as movement from point to point, is necessary if we want to compute trajectories it gives rise to paradoxes when taken as more than a mathematical device. Is this not the case?

 

[matt grime]

How about you read this:

http://www.dpmms.cam.ac.uk/~wtg10/reals.html

 

[Hurkyl]

it gives rise to paradoxes

What gives rise to paradoxes? And of what paradoxes do you speak?

 

[Canute]

The best known are of course Zeno's. And before everybody piles into say that the calculus or something else solves these paradoxes, they are still outstanding. Thus physicist Peter Lynds cites them in his recently published papers on the incoherence of the concept of 'instants' of time. I share his view. (They're available at the Cern site I think).

But I've got to stop here, really. I'm away for a fortnight and so trying to extricate myself from various discussions. Sorry to dive out just when things were going to hot up. Thanks for the discusion, I've a better understanding of the issues than when it started. I might risk a question on Zeno when I get back if I'm feeling brave enough.

Bye for now

Canute

 

[Hurkyl]

I usually try not to have the last word before leaving a discussion.

I rather get the impression that these objections arise by forgetting the notion of topology: you have a neighborhood structure in addition to the set of points.

 

[Canute]

You're quite right. I happened to notice your reply before I logged off. I didn't mean to just have the last word and run off, sorry if that was the impression. I was just explaining my comment about the paradoxicality of motion if spacetime is treated as series of points a la Zeno, which you'd asked about, and explaining that I was about to disappear from the discussion. I didn't want to go without warning. I shouldn't have mentioned Lynds. I'd like to discuss Zeno with you but will have to come back another time. Sorry if it seemed I was point scoring. My mistake.

Regards
Canute

 

[HallsofIvy]

Dedekind's axiom doesn't have anything to do with "motion" or Zeno. It is equivalent to the least upper bound property (or "monotone convergence", or the "Cauchy criterion" or the fact that the set of real numbers is connected (with the usual topology) or that every closed and bounded set of real numbers is compact (again with the usual topology).

 

[mathwonk]

you are not to blame. Dantzig's axiom is poorly worded since as stated it did not define the word "severing".

a more precise statement would have been: "... then either the first class has a largest element or the second class has a smallest one."

Unless this is a very good book for other reasons, I would try to find one that is written more carefully.

 

[Canute]

Aha. That makes sense. It hadn't occurred to me that Dantzig might have misrepresented D's axiom. Thanks.