1. what is the smallest distance between two points measurable?

2.

3. Originally Posted by parag1973
what is the smallest distance between two points measurable?
What do you mean by measureable ?

This does not sound like a mathematics question but rather a question in engineering or physics. It is possible to measure points using interfrence patterns from light that are quite small. It is possible to make images of objects smaller than the wavelenght of visible light -- using an electron microscope.

4. Zero.

In the event that the term "distance" makes sense, this implies a metric, say . One, just one, of the conditions that a metric must satisfy is that .

Or, if you prefer, implies . Another condition our metric requires is that the value is either definitely positive, or definitely zero. Mathematicians use the highly unlovable expression "positive-definite" for this condition.

You mean, I suppose, that given and is a positive real number or zero, what is the smallest such non-zero number where we can be sure that .

To answer this, you need to know what is the least non-zero positive real number. I have no idea, and I am not alone. All agree (by the well-ordering theorem) that there is one, but nobody knows what it is.

Confused? You should be. So are we all!

I am teasing, of course. You are probably asking about something physical, like Planck length. If so, I suggest you pop "downstairs" and ask a physics jock

5. I moved this reply to the more appropriate "general" forum

6. Originally Posted by Guitarist
...

To answer this, you need to know what is the least non-zero positive real number. I have no idea, and I am not alone. All agree (by the well-ordering theorem) that there is one, but nobody knows what it is.

Confused? You should be. So are we all!

...
I am certainly confused. If x were the smallest positive real number then what might x/2 be ?

The real numbers are not well-ordered.

7. It seems that Guitarist is still confused about the well-ordering principle. I remember serpicojr pointing this out once, but he is still making the same mistake.

Guitarist: The well-ordering principle only says that every nonempty set can be well ordered. It does not state that every ordering is a well order. That is the confusion I believe you are still harbouring.

What the WOP says in relation to the real numbers is that there exists some ordering of the real numbers under which the real numbers are well ordered. Whatever this ordering may be, it is certainly not the usual ordering of the real numbers. The usual ordering of the real numbers is not a well order.

8. Originally Posted by JaneBennet
It seems that Guitarist is still confused about the well-ordering principle. I remember serpicojr pointing this out once, but he is still making the same mistake.

Guitarist: The well-ordering principle only says that every nonempty set can be well ordered. It does not state that every ordering is a well order. That is the confusion I believe you are still harbouring.

What the WOP says in relation to the real numbers is that there exists some ordering of the real numbers under which the real numbers are well ordered. Whatever this ordering may be, it is certainly not the usual ordering of the real numbers. The usual ordering of the real numbers is not a well order.
OK if that is the source of the confusion I begin to see the problem.

The well-ordering principle like Zorns's Lemma and The Hausdorff Maximal Principle is a consequence of the axiom of choice that is sometimes used for what is known as transfinite induction. This is a very useful device and without it a good portion of modern mathematics could not proceed. But if the Well-ordering Pinciple is creating confusion, then forget all about it and use one of the other versions -- in applications they are pretty much interchangeable.

9. Jane, please don't patronize me.

I never asserted that the usual (total) ordering on R is a well-order.

Had I, for example, called my metric a "distance function", or, equivalently, said it must satisfy the triangle inequality, the substance, if not the tone, of your remarks might have had more merit. I did neither of these things. My only use of the term "distance" was written just like that - in quotes.

10. Patronize you? I was merely pointing out a mistake of yours that had already been pointed out once before to you – but which, apparently, you had not taken on board!

http://www.thescienceforum.com/viewt...=133833#133833

Originally Posted by serpicojr
Originally Posted by Guitarist
First we have, among many, the Well-Ordering Theorem: any set can be well-ordered, that is, any subset of a set has a least element. Try and apply this to the real line . The theorem asserts that there is an element in (0, 1) for which there is no smaller element.
Careful there. The theorem implies any set can be well-ordered, not that any order is a well order. So you can define an order on, say, under which every subset has a least element. However, many orders that are not well-orders exist--for example, the usual order on . has no least element under this, as given any element , and .
The fact that you were still not quite clear on this is borne out by the bold part of the quoted portion of your post below.

Originally Posted by Guitarist
You mean, I suppose, that given and is a positive real number or zero, what is the smallest such non-zero number where we can be sure that .

To answer this, you need to know what is the least non-zero positive real number. I have no idea, and I am not alone. All agree (by the well-ordering theorem) that there is one, but nobody knows what it is.
Another point is that the WOP only says that the reals can be well ordered by some order – it does not say that the reals under this well-order is a metric space! You started by taking a metric space, and then trying to apply the WOP (which has nothing to do with metric spaces whatsoever) to your metric space. I’m not sure that this is a valid move.

11. Originally Posted by JaneBennet
Another point is that the WOP only says that the reals can be well ordered by some order – it does not say that the reals under this well-order is a metric space!
I assume you are talking under the order topology induced by the well ordering. I think that this question could have something to do with the continuum hypothesis (at least the real line is first countable under this order topology if CH is true). Ah, but it is not paracompact if CH is true and thus not metrizable

William please stop these offensive digs at Jane. Thank you.

I think you just made the only insult to her. I doubt she needs your help.

Sincerely,

William McCormick

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