Or is it infinitely small? Prince James has been insisting that points (in any coordinate system) have "mathematical size". I'm educated as an engineer, and my understanding of mathematical principles does not allow a point any "mathematical size", but simply a location of size zero.

I have argued fairly extensively that this is simply wrong, and that points cannot have size by definition. College has been a while ago though and I'm unsure if there's something I'm overlooking entirely.

Please help.

Hi Wes,

I don't know what's been said or what Prince James means by "mathematical size", so I obviously can't comment about previous discussions.

That being said, I think there is a certain natural sense in which a point has zero size. Consider a point x in one dimension. We can construct a sequence of closed intervals around this point with ever decreasing size. For example, consider intervals of the form [x - 1/n,x+1/n] where n is an integer. You can easily see that the length or size of such an interval is 2/n. Now, x is contained in each of these intervals, so with any reasonable notion of size we should have that the size of our point is less than or equal to the size of each interval. This can only be true if the "size" of our point is zero. In other words, a point has no size because it can be contained in an interval of arbitrarily small size.

In the math jargon one says that a point is a set of measure zero.

Hope this helps.

Physics Monkey:

Clarification: I meant mathematic size as Wesmorris thought I meant that all graphs needed a physical analogue. That is to say, I am not arguing that a triangle drawn on a graph need to be analogus to the pyramids in real life, thus they only have size in a mathematical sense.

In regards to your answer in a point, would not your answer also give you the infinitely small? For if the infinitely small is defined as "infinitely distant from any finite point or an infinitely large point", it will always be less (by an infinite amount) than any finite value, but will retain a notion of size which will allow it, in an infinite quantity, fill every finite value and the infinitely large (remembering that all other points are infinitely distant from it).

Also, if a point is zero in size, and any line can be divided into points, then would it not be impossible to ever reach the line? For a size of zero, even over an infinite timespan of multiplicaiton, could never go beyond zero.

Also, if a point is zero in size, and any line can be divided into points, then would it not be impossible to ever reach the line? For a size of zero, even over an infinite timespan of multiplicaiton, could never go beyond zero.

No, 0\times\infty is undefined. If a point had non-zero size, then a line would correspond to a *finite* number of points, which is obviously wrong.

See also "Lebesgue Measure," which formalizes the usual ideas about the size of objects:

http://en.wikipedia.org/wiki/Lebesgue_measure

Quadraphonics:

0 * infinity = undefined? Since when? Is it not an axiom of multiplication that X * 0 = 0? Why then should this be different for infinity?

If the above is true, then to say that a point with zero size could produce an "infinite amount of points in a line" is wrong, for there would be no points whatsoever. Only the infinitely small could do so, on account of the equality of every finite point and the infinitely large being infinitely distant from the infinitely small.

0 * infinity = undefined? Since when?

Since always.


Is it not an axiom of multiplication that X * 0 = 0?


For any X \neq \infty, yes.


Why then should this be different for infinity?


Because infinity is not a real number. It's a notation for describing the behavior of limits. Many of the usual axioms of arithmetic do not apply to infinity. For example, X - X = 0 does not hold; \infty - \infty is undefined.

More here:

http://en.wikipedia.org/wiki/Infinity#Mathematical_infinity


If the above is true, then to say that a point with zero size could produce an "infinite amount of points in a line" is wrong, for there would be no points whatsoever. Only the infinitely small could do so, on account of the equality of every finite point and the infinitely large being infinitely distant from the infinitely small.

I have no idea what this statement is intended to mean.

Hi Wes,

I don't know what's been said or what Prince James means by "mathematical size", so I obviously can't comment about previous discussions.

That being said, I think there is a certain natural sense in which a point has zero size. Consider a point x in one dimension. We can construct a sequence of closed intervals around this point with ever decreasing size. For example, consider intervals of the form [x - 1/n,x+1/n] where n is an integer. You can easily see that the length or size of such an interval is 2/n. Now, x is contained in each of these intervals, so with any reasonable notion of size we should have that the size of our point is less than or equal to the size of each interval. This can only be true if the "size" of our point is zero. In other words, a point has no size because it can be contained in an interval of arbitrarily small size.

In the math jargon one says that a point is a set of measure zero.

Hope this helps.

Well put, thank you. Mathematically, is there such a thing as "the infinitely small", and does it apply to the notion of "mathematical size" in the way you understand Prince James to mean it as witnessed thus far in this thread?

http://en.wikipedia.org/wiki/Infinitesimal

"Nonzero infinitesimals are not members of the set of real numbers"

So if the graph in question is inclusive of only real numbers, then there is no such thing as an infinitesimal on it.

Quadraphonics:

Since always.

For any , yes.

I just looked that up. You are correct, although I find it remarkably silly that such is held to be so, considering that infinity has properties which would admit of invalidating the principle that X * 0 = 0. One can make at least a decent (although I am of the opinion deeply flawed) defense of 0^0 = 1, and so counting it as defined is not as odious as counting this as undefined.

Because infinity is not a real number. It's a notation for describing the behavior of limits. Many of the usual axioms of arithmetic do not apply to infinity. For example, does not hold; is undefined.

Again, I would critique that, but yes, understood that this is current math theory.

I have no idea what this statement is intended to mean.

Here's my proof given simply.

Let IS = Infinitely small. IL = Infinitely large.

All finite numbers and IL are defined as infinitely distant from IS and vice-versa for IL.

Now it is rather clear that x * Infinity = IL (this is even shown on your Wiki page you linked, by the way). Similarly, even if x = IS, x * infinity = IL. Yet as all finite numbers and IL are equidistant from IS, than the answer given from x * infinity = IL, stands for all finite numbers, and accordingly, an infinite amount of IS points can be found in any line of any finite value.

On the other hand, a zero sized point could never compose a line. Either it would be indeterminate if you take the orthodox view of contemporary mathematics, or if it follows the axioms of finite mathematics or is simply finitely multiplied, it could not compose the line as it would never reach the value of the line through multiplication. Moreover, it does not at all stand to reason that something with length (or any other dimensional quality) could be composed of that which has none.

I've been arguing this with Swivel and somewhat with Wesmorris over in the philosophy section, hence why he brought all this up.

Prince James...

Look at lim(x->oo) 1/x*x = 1.

Now, look at lim(x->oo) 1/x^2 * x = 0

Both are just like oo*0, but we have two different answers.

Mathematically, is there such a thing as "the infinitely small", and does it apply to the notion of "mathematical size" in the way you understand Prince James to mean it as witnessed thus far in this thread?

The smallest thing is typically zero. Most mathematical notions of size (or area or volume) consist of non-negative numbers, and so the smallest size is zero. As Physics Monkey's earlier example illustrates, this can be thought of as "infinitely small" in the sense that \lim_{x \to \infty}\frac{1}{x} = 0.


"Nonzero infinitesimals are not members of the set of real numbers"

So if the graph in question is inclusive of only real numbers, then there is no such thing as an infinitesimal on it.

Except zero, yeah.

Here's my proof given simply.

Let IS = Infinitely small. IL = Infinitely large.

I.e., IS = 0 and IL = infinity.


All finite numbers and IL are defined as infinitely distant from IS and vice-versa for IL.

No, this won't work. If X \in \mathbb{R}, then \infty - X = \infty but X - 0 = X. You can't be infinitely far from both infinity and zero at the same time, and finite numbers can't be infinitely far from zero by definition.

Here's my proof given simply.

Let IS = Infinitely small. IL = Infinitely large.

All finite numbers and IL are defined as infinitely distant from IS and vice-versa for IL.
If IS is in the vicinity of zero, why would a finite number minus IS be infinite?
Now it is rather clear that x * Infinity = IL (this is even shown on your Wiki page you linked, by the way).
Okay, if x > 0, and 'going along' with your less-than-rigourous notion of "infinitely large".
Similarly, even if x = IS, x * infinity = IL.
If you mean something like \left( \lim_{ \epsilon \rightarrow 0} \: \epsilon \right) \times \infty = \infty, then all right, provided we accept x \times \infty = \infty for x > 0, and take the limit with \epsilon > 0.
Yet as all finite numbers and IL are equidistant from IS,
This makes no sense.
than the answer given from x * infinity = IL, stands for all finite numbers.
...therefore 0 \times \infty = \infty?

0 \times \infty is left undefined because, just as with \frac 0 0, different "derivations" of it yield different results. See Absane's post on this.
On the other hand, a zero sized point could never compose a line.
All you've really offered to support this so far is your intuition. Others have shown obvious problems that arise if a point is assigned a non-zero measure; zero's really all that's left.

Actually this reminds me of the subtle difference between "impossible" and "probability zero" events in probability theory. For example, if you are asked to pick a real number at random in the interval [0,1[, what's the probability that you'll pick a given number (eg. 0.5) in the interval?

I suppose the idea of different "orders" of zero (analagous to the different orders of infinity already defined in mathematics) might be fun to play around with a bit.

The smallest thing is typically zero. Most mathematical notions of size (or area or volume) consist of non-negative numbers, and so the smallest size is zero. As Physics Monkey's earlier example illustrates, this can be thought of as "infinitely small" in the sense that \lim_{x \to \infty}\frac{1}{x} = 0.

Pretty much what I said in the other thread. Thanks for the sanity, Wes.

Also, if a point is zero in size, and any line can be divided into points, then would it not be impossible to ever reach the line?

The two are not defined across each other in this particular way. They are discrete concepts. A point is an infinitesimal volume in space, a line has infinitesimal area, and infinite length, a plane has infinitesimal thickness, and so on.

This is much like an old Greek riddle on the tortoise and the hare (maybe it's not Greek) - how when the hare catches up to the tortoise, the tortoise moves forward a bit, and by the time the hare reaches that point, the tortoise moves ahead just a bit further still, ad infinitum, ad nauseum. It's a good start into the concept of infinity and infinitesimals.

Isn't that zenu's paradox?

Wesmorris:

Zeno. Zenu is the guy from Scientology. ;)

Absane:

Look at lim(x->oo) 1/x*x = 1.

Now, look at lim(x->oo) 1/x^2 * x = 0

If the limit of x is approaching infinity, why would the first equal one? Through a simple reversal of the division?

Why would also x^2 produce zero in the next case?

Absane:



If the limit of x is approaching infinity, why would the first equal one? Through a simple reversal of the division?

Why would also x^2 produce zero in the next case?

lim(x->oo) 1/x*x = x/x = 1

Next.

lim(x->oo)1/x^2 * x = x/x^2 = 1/x = "1/oo" = 0

Quadraphonics:

Can it be properly considered that zero can be infinitely small? For does not zero imply a lack of size, geometrically speaking, rather than a smallness of such?

But yes, infintesimals are not part of the real numbers.

I.e., IS = 0 and IL = infinity.

No - not zero. I claim there is a difference, on account of the the fact that zero * IL = undefined in orthodox mathematics. This is unacceptable, as well as the fact that processes of division never reduce to zero.

No, this won't work. If , then but . You can't be infinitely far from both infinity and zero at the same time, and finite numbers can't be infinitely far from zero by definition.

This would be very true if it were zero, but I am not claiming it is so. But yes, I'll certainly admit of your critique for zero and concede on that, though that point I am not making at all.

This is where we should have been having this discussion in the first place.

I learned something wonderful in this thread: The use of oo to symbolize infinity! Glorious.

Heh.. I started using oo a long time ago because it's quick, easy, and I am too lazy to get the correct symbol in there.

Can it be properly considered that zero can be infinitely small? For does not zero imply a lack of size, geometrically speaking, rather than a smallness of such?

No, zero is a perfectly well-defined size. I can't imagine what you could possibly mean by "infinitely small" other than "approaching zero." If something has a size greater than zero, it cannot be "infinitely small" in any meaningful sense, because I can easily come up with infinitely many numbers that are smaller than it is.

przyk:

If IS is in the vicinity of zero, why would a finite number minus IS be infinite?

It would not. I never claimed this?

I said that any finite number would be equally distant from both extremes of infinity, not that if one took away the infinitely small from any number that such a number would be infinite.

If IS is in the vicinity of zero, why would a finite number minus IS be infinite?

Yes, I should have noted no 0. Sorry about that. Non-zero finite number or IL.

If you mean something like , then all right, provided we accept for , and take the limit with .

That seems about right (although sorry for not pasting your notation as it won't let me paste it evidently).

This makes no sense.

How do you figure?

If something is infinitely small, is not any finite value infinitely distant from it?

And if something is infinitely large, is not any finite value infinitely distant from it?

Non-zero, mind you.

...therefore ?

is left undefined because, just as with , different "derivations" of it yield different results. See Absane's post on this.

It would certainly be true if IS = 0. IS is properly the lowest non-zero value, but evidently I did not make this clear enough in my critique of points (and infintesimals) as zero.

But anyway, onward.

All you've really offered to support this so far is your intuition. Others have shown obvious problems that arise if a point is assigned a non-zero measure; zero's really all that's left.

Actually, most assumed I meant IS = zero. But this wasn't the case.

I will have shown more than "intuition" if you look over my post now that it is different.

Actually this reminds me of the subtle difference between "impossible" and "probability zero" events in probability theory. For example, if you are asked to pick a real number at random in the interval , what's the probability that you'll pick a given number (eg. 0.5) in the interval?

I suppose the idea of different "orders" of zero (analagous to the different orders of infinity already defined in mathematics) might be fun to play around with a bit.

That would be interesting to discuss, yes.

Facial:

The two are not defined across each other in this particular way. They are discrete concepts. A point is an infinitesimal volume in space, a line has infinitesimal area, and infinite length, a plane has infinitesimal thickness, and so on.

Infinitesimal as in zero, or non-zero?

Also, do not the dimensions build upon one another? A line is built of points. A plane built of lines. A 3d object of planes...

This is much like an old Greek riddle on the tortoise and the hare (maybe it's not Greek) - how when the hare catches up to the tortoise, the tortoise moves forward a bit, and by the time the hare reaches that point, the tortoise moves ahead just a bit further still, ad infinitum, ad nauseum. It's a good start into the concept of infinity and infinitesimals.

Yes. Zeno's paradox of Achilles and the Tortoise is excellent.

Absane:

lim(x->oo) 1/x*x = x/x = 1

Next.

lim(x->oo)1/x^2 * x = x/x^2 = 1/x = "1/oo" = 0

Danke.

But okay, tell me if I am making some mistake of comprehension here, just to make sure we are on the same page.

1/infinity (technically a number approaching the limit of infinity) * infinity = infinity/infinity = 1.

Yet would that be the case? For would not infinity be also able to be 2, 5, 121341, or any other number? And if the lowest amount would be ideal to pick, wouldn't it be best to pick say, an infintesimal?

In regards to the second, if x is a value approaching infinity, x^2 works, but if we take x as infinity, it would not (as properly speaking infinity^X = Simply infinity).

And how can 1/infinity = zero? Zero * infinity = Undefined and/or zero (I say zero...orthodox mathematics says undefined).

I said that any finite number would be equally distant from both extremes of infinity, not that if one took away the infinitely small from any number that such a number would be infinite.

But the two statements are equivalent. The distance between two numbers is the magnitude of their difference. So to say that a finite number is infinitely distant from IS is exactly the same as saying X - 0 = \infty, which contradicts the assumption that X is finite.


It would certainly be true if IS = 0. IS is properly the lowest non-zero value, but evidently I did not make this clear enough in my critique of points (and infintesimals) as zero.

There is no "lowest non-zero value," at least when it comes to real numbers. For any non-zero value you care to suggest, I can immediately construct a smaller one by dividing it by 2. IS, as you have defined it, must be zero.

Quadraphonics:

No, zero is a perfectly well-defined size. I can't imagine what you could possibly mean by "infinitely small" other than "approaching zero." If something has a size greater than zero, it cannot be "infinitely small" in any meaningful sense, because I can easily come up with infinitely many numbers that are smaller than it is.

What about the ideal value of a process of infinite division? Would not this count as an infinitely small that you could not provide an answer that is lower for?

What about the ideal value of a process of infinite division?

You'll have to clarify what you mean by this. I don't know what you mean by
"a process of infinite division," nor the "ideal value" that correspond to it.

Quadraphonics:

Say a process of division beginning at one and divided by two ad infinitum.

1/2 = 1/2 / 2 = 1/4...1/8...16...32...64...128...

Do we say that a plane is a three-dimensional object whose thickness is zero? No. We say it's a two-dimensional object.

Do we say that a line is a two-dimensional object whose width is zero? No. We say it's a one-dimensional object.

So why are we trying to say that a point is a one-dimensional object whose length is zero?

A point is a zero-dimensional object. It has location but no size.

But the two statements are equivalent. The distance between two numbers is the magnitude of their difference. So to say that a finite number is infinitely distant from IS is exactly the same as saying X - 0 = \infty, which contradicts the assumption that X is finite.



There is no "lowest non-zero value," at least when it comes to real numbers. For any non-zero value you care to suggest, I can immediately construct a smaller one by dividing it by 2. IS, as you have defined it, must be zero.

PJ, this is what I mean when I say that you want your infinitesimal to act like Zero, but you want to pretend that it has a non-zero value. It was also the point that I tried to make by showing you that your IS could lead to IS/2, which would be the new IS, ad infinitum. You can see how the Law of Identity is an escape clause here.

Quadraphonics:

Say a process of division beginning at one and divided by two ad infinitum.

1/2 = 1/2 / 2 = 1/4...1/8...16...32...64...128...

And here is the problem of having a dynamic process, then later treating it as if it is a static value. The Eternity of the process does not equate a fulfilled Infinity of smallness.

Remember the D*shes? Here they are again. Pick one, any one, if they are non-zero they all serve the exact same purpose.

Say a process of division beginning at one and divided by two ad infinitum.

1/2 = 1/2 / 2 = 1/4...1/8...16...32...64...128...

Okay. This is the sequence 2^{-n}, where n is the number of divisions by two that have been performed. I assume that by "ideal value," you are referring to where you'd end up if you kept doing divisions forever. I.e.:

\lim_{n \to \infty} 2^{-n}

As it happens, this limit is easy to compute, and is equal to 0. So this approach does not produce a non-zero IS, but rather demonstrates the fact that IS must be 0.

Do we say that a plane is a three-dimensional object whose thickness is zero?

Sometimes. It depends on whether we're interested in volume or surface area. This is all driven by the choice of measure. There are measures which assign non-zero size to singletons (counting measure, for example), but they are not appropriate for the sorts of geometric considerations in this thread. Lebesgue measure is the appropriate thing to use here, and it certainly assigns zero size to single points (or, indeed, any countable union of singletons).

Swivel:

You can see how the Law of Identity is an escape clause here.

When the very next statement contradicts the prior, it is most certainly not an "escape clause" to hold up the violation of Law of Identity.

And here is the problem of having a dynamic process, then later treating it as if it is a static value. The Eternity of the process does not equate a fulfilled Infinity of smallness.

Remember the D*shes? Here they are again. Pick one, any one, if they are non-zero they all serve the exact same purpose.

And again, your lack of consideration for the "metagenie" and the ideal value of the process undercuts your points.

The ideal value of any process of eternal division of a non-zero number, assuredly would have to be "infinitely small".

Quadraphonics:

Okay. This is the sequence , where is the number of divisions by two that have been performed. I assume that by "ideal value," you are referring to where you'd end up if you kept doing divisions forever. I.e.:



As it happens, this limit is easy to compute, and is equal to 0. So this approach does not produce a non-zero IS, but rather demonstrates the fact that IS must be 0.

Might you explain why it is zero?

For wouldn't this be impossible to demonstrate, if zero * infinity = undefined, as has been claimed? For the process would not be reversable. Also, since when can division produce zero?

Might you explain why it is zero?

Let \epsilon be any positive real number. Then:

2^{-n} \leq \epsilon \,\, , \, \forall n \geq n_0 = \lceil -\log_2(\epsilon)\rceil

So, because the sequence will be less than any positive number \epsilon after some finite number of divisions n_0, it follows that the limit of the sequence must be less than any positive real number. Since the sequence is also non-negative, it follows that the limit is zero.


For wouldn't this be impossible to demonstrate, if zero * infinity = undefined, as has been claimed?

As you can see, the proof does not require one to perform the arithmetic operation 0 \times \infty.


For the process would not be reversable.

It doesn't make sense to talk about reversing a process that never ends.


Also, since when can division produce zero?

A finite number of divisions cannot produce zero. But if you keep doing divisions forever, you'll end up with zero.

Well put, thank you. Mathematically, is there such a thing as "the infinitely small", and does it apply to the notion of "mathematical size" in the way you understand Prince James to mean it as witnessed thus far in this thread?

Some history: Both Newton and Leibniz used the fuzzy concept of an infinitesimally small but non-zero quantity in their developments of the calculus. These concepts were not well defined. Mathematicians discarded those nonrigorous concepts when Weierstrass rigorously developed the concept of a limit. Physicists, on the other hand, kept on using \Delta x and its ilk because the limit concept is a bit tedious and harder to grasp.

Fast-forward to the 1960s: Robinson's nonstandard analysis gave rigor to the infinitesimal. However, adding rigor makes the infinitesimals even more tedious than the epsilon-delta limit stuff that introductory calculus students fear and loathe.

One key point about the non-standard analysis: It subsumes standard analysis. Any result that is true in standard analysis is also true in the non-standard analysis. A point has zero size.

Do we say that a plane is a three-dimensional object whose thickness is zero? No. We say it's a two-dimensional object.

I would say no depending on how you define 2D.

If you know anything about subspaces, you could see that R^{2} is not a subspace of R^{3}, yet you can define an infinate number of planes in R^{3}. But this is a consequence of definitions.

If you know anything about subspaces, you could see that R^{2} is not a subspace of R^{3} ...

{\mathbb R}^2 most definitely is a subspace of {\mathbb R}^3.

http://en.wikipedia.org/wiki/Linear_subspace
Example I: Let the field K be the set b]R[/b] of real numbers, and let the vector spaceV be the Euclidean space R3. Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.

This example shows {\mathbb R}^2 is a subspace of {\mathbb R}^3. More from the article:
In general, any subset of an Euclidean space Rn that is defined by a system of homogeneous linear equations will yield a subspace. (The equation in example I was z = 0.) Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.

{\mathbb R}^2 most definitely is a subspace of {\mathbb R}^3.

http://en.wikipedia.org/wiki/Linear_subspace
Example I: Let the field K be the set b]R[/b] of real numbers, and let the vector spaceV be the Euclidean space R3. Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V.

This example shows {\mathbb R}^2 is a subspace of {\mathbb R}^3. More from the article:
In general, any subset of an Euclidean space Rn that is defined by a system of homogeneous linear equations will yield a subspace. (The equation in example I was z = 0.) Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.

Of course, (x,y,0) is a subspace of R^3, but (x, y) doesn't make sense in R^3 without a z component.

http://www.physicsforums.com/showthread.php?t=129661

Look at Shmoe's post.

quadraphonics:

So, because the sequence will be less than any positive number after some finite number of divisions , it follows that the limit of the sequence must be less than any positive real number. Since the sequence is also non-negative, it follows that the limit is zero.

All granted...but is not the infinitesimal a hyperreal number to begin with? Specifically a non-zero hyperreal?

All the above would prove is just that it is not a real number. Considering its assumed properties to begin with, this isn't even surprising.

It doesn't make sense to talk about reversing a process that never ends.

Assuming we're dealing with the end result as a whole, it does. No?

A finite number of divisions cannot produce zero. But if you keep doing divisions forever, you'll end up with zero.

Or a hyperreal, it seems.

D.H.:

Here is a straight forward question: If it is held that a point has no dimension and no size, whatever meaningful position can it have in geometry? It cannot rightfully even stand in as the foundation for the first dimension.

Here is a straight forward question: If it is held that a point has no dimension and no size, whatever meaningful position can it have in geometry?

The one and only meaningful position a point can have in geometry is a meaningful position.

It cannot rightfully even stand in as the foundation for the first dimension.

Why would you think that it should? The plane, line, and point are axiomatic in geometry.

In fact, isn't a point lacking a definition simply because there is no mathematics before to describe a point? Much like trying to prove an axiom... it's impossible.

Off topic: 0 \neq 1 is an axiom, correct?

Off topic: 0 \neq 1 is an axiom, correct?

Not quite. Zero is defined as the ordinality of the null set, 1 as the ordinality of the set {0}. In ZFC, it is the axiom of extensionality\forall A, \forall B: A = B \Leftrightarrow (\forall C : C \in A \Leftrightarrow C \in B) that says that the null set is not equal to the set {0}.

All granted...but is not the infinitesimal a hyperreal number to begin with? Specifically a non-zero hyperreal?
All the above would prove is just that it is not a real number.

No, the real numbers are complete, so we know that the limit of the sequence is also a real number, and hence is zero.


Or a hyperreal, it seems.


If you want to work in terms of nonstandard analysis, that's fine, but you should do it correctly. You certainly cannot construct a hyperreal number as the limit of a sequence of real numbers. You need to introduce ultrafilters and so on. At any rate, your basic mistake is just as obvious in terms of hyperreals, which is to say that the distance from an infintesimal to any finite number is finite, not infinite.

PJ, why are you so wed to this idea? Are you worried that god becomes impossible in light of these revelations? You seem to be struggling to maintain an incorrect position, which usually means ulterior motives of some sort. I'm honestly curious about this.

Not quite. Zero is defined as the ordinality of the null set, 1 as the ordinality of the set {0}. In ZFC, it is the axiom of extensionality\forall A, \forall B: A = B \Leftrightarrow (\forall C : C \in A \Leftrightarrow C \in B) that says that the null set is not equal to the set {0}.

Where can I read about this? I am very curious about what the "official" axioms of mathematics are. Or do we not really have this? Anything I read about "axioms" is usually just something simple like I have stated... I'm looking for everything put into real mathematical notation (like that you supplied). Since I know how to read language like \forall A, \forall B: A = B \Leftrightarrow (\forall C : C \in A \Leftrightarrow C \in B), I am quite disappointed that I am not taught anything in that language. In fact, I am only one of a few math students at my school that formalizes proofs with this.

Where can I read about this? I am very curious about what the "official" axioms of mathematics are. Or do we not really have this? Anything I read about "axioms" is usually just something simple like I have stated... I'm looking for everything put into real mathematical notation (like that you supplied). Since I know how to read language like \forall A, \forall B: A = B \Leftrightarrow (\forall C : C \in A \Leftrightarrow C \in B), I am quite disappointed that I am not taught anything in that language. In fact, I am only one of a few math students at my school that formalizes proofs with this.

There are no "official" axioms of mathematics. For example, replacing the parallel postulate with something else leads to very interesting and useful mathematics. That said, more-or-less standard sets of axioms for number theory and set theory do exist: the Peano postulates (number theory) and the Zermelo–Fraenkel axioms plus the Axiom of Choice, aka ZFC (set theory). As a starter, Mathworld (http://mathworld.wolfram.com) and Wikipedia have fairly good articles on number theory and set theory.

D H:

The one and only meaningful position a point can have in geometry is a meaningful position.

How can one have position without space or dimension? This seems, at best, a half-conceived notion.

Damn you, Euclid, for not being more rigorous!

Why would you think that it should? The plane, line, and point are axiomatic in geometry.

Is not a line defined as extending between two points?

quadraphonics:

No, the real numbers are complete, so we know that the limit of the sequence is also a real number, and hence is zero.

That seems contradictory to the definition of an infinitesimal as presented in modern mathematics.

http://mathforum.org/dr.math/faq/analysis_hyperreals.html

If you want to work in terms of nonstandard analysis, that's fine, but you should do it correctly. You certainly cannot construct a hyperreal number as the limit of a sequence of real numbers. You need to introduce ultrafilters and so on. At any rate, your basic mistake is just as obvious in terms of hyperreals, which is to say that the distance from an infintesimal to any finite number is finite, not infinite.

How so? If the infinitely small is defined as always less than any positive, non-zero, real number, then it would stand to reason that it is infinitely distant from all, would it not?

Swivel:

PJ, why are you so wed to this idea? Are you worried that god becomes impossible in light of these revelations? You seem to be struggling to maintain an incorrect position, which usually means ulterior motives of some sort. I'm honestly curious about this.

When have I ever brought God into this?

And no, I am hardly trying to maintain an erroneous position: I simply am ill convinced that my position is as such.

In fact, it remains to me quite manifest that I am right, specifically in light of the hyper real definition of an infinitesimal. As you and I are both inclined to not accept mere "arguments from authority" as you have spoken about in the other thread, of course there is much to debate here.

But then again, if our good fellows can help prove your case and it seems to me that I am wrong, I'll gladly concede.

D H:

How can one have position without space or dimension? This seems, at best, a half-conceived notion.

Explain why you think this way. It makes perfect sense to me.

[/QUOTE]Is not a line defined as extending between two points?[/QUOTE]

Yes, it is. But what exactly constitutes a line is not defined; it is axiomatic.


quadraphonics:

That seems contradictory to the definition of an infinitesimal as presented in modern mathematics.

http://mathforum.org/dr.math/faq/analysis_hyperreals.html


The reals are the complete ordered field (i.e., any other field that is complete and ordered is isomorphic to the reals).



How so? If the infinitely small is defined as always less than any positive, non-zero, real number, then it would stand to reason that it is infinitely distant from all, would it not?[/QUOTE]

No. Think about it this way: The difference between the real numbers 0 and 1 is 1. The distance between 0 + some infinitesimal quantity and 1 is infinitesimally smaller than 1. It is not infinite.

That seems contradictory to the definition of an infinitesimal as presented in modern mathematics.

No, the hyperreals do not correspond to limits of sequences of real numbers. Rather, they encode both the limit *and* how quickly the sequence approaches the limit. This is what allows us to rank-order them, and use them to define the products and quotients of infintesimals and other quantities. For example, the sequences {1, 1/2, 1/4, ....} and {1, 1/3, 1/9, ...} both have 0 as their limit, but correspond to distinct hyperreal numbers (because the former is always greater than the latter). That is, the "standard part" of both hyperreal numbers is 0 (the common limit of the sequences), but the infintesimal parts differ (reflecting the different convergence rates).


How so? If the infinitely small is defined as always less than any positive, non-zero, real number, then it would stand to reason that it is infinitely distant from all, would it not?

No. For example, zero is an infintesimal, and the distance from 0 to 1 is 1, not infinity (as DH mentioned). The distance from any infinstesimal to any finite hyperreal is itself finite. It sounds as though you are confusing distance (which is just the magnitude of the difference of two numbers) with some concept of "size." That is, non-zero numbers are infinitely *larger* than all infintesimals, in the sense that you have to multiply an infintesimal by an infinite number to get a finite result. Likewise, 0 is infinitely smaller than 1 in the sense that 0 times any finite number is still less than 1. But that doesn't mean the *distance* from 0 to 1 is infinite.

To summarize:

The distance from any infintesimal to any finite number is finite.
The distance from any infinite number to any finite (or infintesimal) number is infinite.

Any infinite number is infinitely larger than any finite number (or infintesimal).
Any finite number is infinitely larger than any infintesimal.

So, you see, distance and size are not the same thing.

But then again, if our good fellows can help prove your case and it seems to me that I am wrong, I'll gladly concede.

No, you won't. Your ideas have been proven misguided a dozen different ways by several different posters, myself included. And you keep repeating the same incorrect things over and over again. You are wed to your idea.

I'm not arguing from authority, nor do I appeal to it. See the other thread for a very simple list of problems with your concept of the infinitesimal. Those arguments stand on their own. The fact that this thread has also demolished your ideas, in several different ways, should hopefully get you re-thinking them.

There are a finite number of "bits" in a finite length of string. The fact that you argue this is bizarre to me.

A point has no size nor dimension. The fact that you argue this is bizarre to me.

There is no non-zero number which acts like zero, the fact that you keep trying to get your infinitesimal to perform these duties is bizarre to me.

So if I often come across a bit snooty, or liken you to a theist, it is only because of the strangeness of your ideas. I know you find this hard to believe, but on this issue you sound like IAC, or LG. Seriously. I'm not trying to ad-hom you here, the merits of my points stand to scrutiny in both threads, I'm just apologizing for any demeaning tone you sense, and explaining its possible origin.


quadraphonics, I really wanted to compliment you on the above post, but since I've already sucked up to you in the economics section, I'm going to abstain. I'm in danger of forming a man-crush.

D H:

Explain why you think this way. It makes perfect sense to me.

A lack of dimension and space makes the notion of position impossible to conceive. In order to properly have a position, one must be able to relate the point back to the rest of the grid. If it has no size, it would presumably not appear on said grid, nor be able to occupy any intersection on said grid.

Yes, it is. But what exactly constitutes a line is not defined; it is axiomatic.

Is not a plane composed of a series of lines?

No. Think about it this way: The difference between the real numbers 0 and 1 is 1. The distance between 0 + some infinitesimal quantity and 1 is infinitesimally smaller than 1. It is not infinite.

I agree. But I was speaking of non-zero reals when I said "infintesimal". Perhaps this was my mistake - not being precise enough.

But consider for a finite, non-zero number, quickly: If the infintesimal is always lower than any finite, non-zero value, then to say that 5 (or any other number) is a finite amount away from the infintesimal implies that the infintesimal is, in fact, not infintesimal as it would be not lower than said finite value. Yet as it is -defined- as such, and accordingly never reachable through a finite process, it can only be claimed to be infinitely distant from any finite vlaue.

A process of infinite division as taken all together would be a reasonable way of discussing a reduction to it.

Edit: In light of Quadraphonics point, I'm retracting my prior statement. It would not be an infinite distance away, it would only take an infinite process of division to reach it, even if the distance would be finite, as the process would resolve to constantly smaller numbers.

quadraphonics:

No, the hyperreals do not correspond to limits of sequences of real numbers. Rather, they encode both the limit *and* how quickly the sequence approaches the limit. This is what allows us to rank-order them, and use them to define the products and quotients of infintesimals and other quantities. For example, the sequences {1, 1/2, 1/4, ....} and {1, 1/3, 1/9, ...} both have 0 as their limit, but correspond to distinct hyperreal numbers (because the former is always greater than the latter). That is, the "standard part" of both hyperreal numbers is 0 (the common limit of the sequences), but the infintesimal parts differ (reflecting the different convergence rates).

So you are saying that the infintesimal represents the "quickness" they resolve to zero?

Yet are not limits supposed to represent the constant movement towards, but never the resolution to, the limit so described? Therefore the answer would never resolve to zero, even over an infinite process, but would rightfully be the infintesimal the process is so described as?

Also, would we not be able to bypass the aforementioned issues regarding multiplication by zero and multiplication of zero by infinity?

No. For example, zero is an infintesimal, and the distance from 0 to 1 is 1, not infinity (as DH mentioned). The distance from any infinstesimal to any finite hyperreal is itself finite. It sounds as though you are confusing distance (which is just the magnitude of the difference of two numbers) with some concept of "size." That is, non-zero numbers are infinitely *larger* than all infintesimals, in the sense that you have to multiply an infintesimal by an infinite number to get a finite result. Likewise, 0 is infinitely smaller than 1 in the sense that 0 times any finite number is still less than 1. But that doesn't mean the *distance* from 0 to 1 is infinite.

You know, I think you're right on this one. The distance would be finite, even if the process would be infinite.

I'll concede that point. I was confusing the length of the process with the notion of an infinite distance to it.

Thanks for helping me see my error in regards to that!

Swivel:

No, you won't. Your ideas have been proven misguided a dozen different ways by several different posters, myself included. And you keep repeating the same incorrect things over and over again. You are wed to your idea.

Considering I just conceded an idea, it seems like you're wrong, my good man.

Also: You have not proven a single one of my ideas wrong. You have attacked the notion of a number larger than zero but lower than any finite value, yet this has been shown to be true even in regards to modern, orthodox mathematics.

We have not dealt with dimension to a conclusion here.

And Quadraphonics here even affirmed one of my arguments of infintesimals requiring a multiplication by infinity in order to reach any finite value:


Quadraphonics wrote:
That is, non-zero numbers are infinitely *larger* than all infintesimals, in the sense that you have to multiply an infintesimal by an infinite number to get a finite result.

I'm not arguing from authority, nor do I appeal to it. See the other thread for a very simple list of problems with your concept of the infinitesimal. Those arguments stand on their own. The fact that this thread has also demolished your ideas, in several different ways, should hopefully get you re-thinking them.

Actually, your arguments were pretty badly constructed on several levels and so far only have proven me wrong on only one mistaken point, which I have since agreed on.

Also Quadraphonics and modern mathematics, evidently, agree with me on two points which you have contested:

1. The aforementioned infinitesimal * infinity = finite value. Aka: All finite values can be represented through an infinite series of infinitely small parts.

2. That infinitesimals are in fact, lower than any finite, non-zero, positive real-number. Aka: They are indivisble and can only be approached through an infinite process.

However, if you have ever made any argument that an infinite distance does not separate the infinitely small from any finite positive real number value, I will agree with you: That was in error.

Also, my proof for an infinity in a finite length requires a reevaluation based on the infinity of the process and the absolute value of infitesimals, as opposed to an equality of distance.

There are a finite number of "bits" in a finite length of string. The fact that you argue this is bizarre to me.

Incorrect, evidently!

A point has no size nor dimension. The fact that you argue this is bizarre to me.

Let it remain bizarre for as long as you wish for it to be.

There is no non-zero number which acts like zero, the fact that you keep trying to get your infinitesimal to perform these duties is bizarre to me.

As I have repeatedly claimed: None of my infinitesimals ever act like zero. I have even shown you where they do not. See the difference between zero * infinitesimal v. infinitesimal * infinity. One is either zero or undefined, whereas the other according to both myself and orthodox mathematics, can amount to any finite value.

So if I often come across a bit snooty, or liken you to a theist, it is only because of the strangeness of your ideas. I know you find this hard to believe, but on this issue you sound like IAC, or LG. Seriously. I'm not trying to ad-hom you here, the merits of my points stand to scrutiny in both threads, I'm just apologizing for any demeaning tone you sense, and explaining its possible origin.

Okay. Understood.

Quadraphonics:

Tell me what you think of this:

x/oo = infinitesimal (infinity borrowing from the other poster who introduced that) where x is any non-zero, non-infinitesimal, real number or infinite number.

The reason for the above being your mentioned "infinitesimal * infinity = any value" whereas if x/oo = 0, 0 * infinity = undefined or 0. Moreover, if it is 0 the process would be irreversible and therefore invalid, as all processes the quotient to any process of division, multiplied by the number one is dividing by, equals the number divided. So for instance, 1 / 2 = .5 * 2 = 1. It also seems remarkably unreasonable to say that any process which if finite would equal zero, would if infinite, equal non-zero.

You can't divide by infinity.

And please tell me how an infinite number of non-zero "bits" can exist within a finite space. You are wrong on this. It would suppose a finite number, which can be multiplied by infinity, and NOT get infinity. Never happen.

And thanks for admitting one of your mistakes. It warms my heart to see you making some progress here. Since you often used as a "proof" your infinite distance scenario, this admission should allow you to discard the rest of your conclusions.

You can't divide by infinity.
No, but you can divide by x, look at the limit as x tends to infinity, and casually refer to this as division by infinity.
And please tell me how an infinite number of non-zero "bits" can exist within a finite space.
Define the intervals
I_n = ] 2^{-(n+1)}, \: 2^{-n} ] , \; n \in \mathbb N
Each I_n has a finite length, and the union of all these intervals is the finite interval ]0, \: 1].

Examples like this one all depend on the different "bits" having different lengths. If you partition a finite space into an infinite number of equally-sized subspaces, their measures must all be zero.

Define the intervals
I_n = ] 2^{-(n+1)}, \: 2^{-n} ] , \; n \in \mathbb N
Each I_n has a finite length, and the union of all these intervals is the finite interval ]0, \: 1].

Examples like this one all depend on the different "bits" having different lengths. If you partition a finite space into an infinite number of equally-sized subspaces, their measures must all be zero.

Exactly. [1/2 + 1/4 + 1/8...] as an infinite set with finite limit. But that is not what PJ claims. This debate, about whether or not a point has zero size, stems from a disagreement PJ and I have been having regarding the nature of his concept of an infinitesimal. He claims that there is a non-zero *fixed* value of which an infinite number can exist within a finite segment.

This confusion arises when PJ imagines dividing that segment into smaller bits. Since he can imagine doing this "forever", with the bits getting smaller and smaller, he imagines that there is an infinite number of these small bits in a line segment, let's say between 0 and 1. My point is that this process may be eternal, but there exists a limit at 0 which prevents you from ever reaching a small number, of which there will be an infinite number of them in that segment of length 1.

PJ feels that there must be such a number and has construed it as: .000...1 Which I have pointed out to him as being an infinite construct somehow bounded on both sides, which is impossible. He has lessened his usage of this construct, but still tries to concoct it by other means.

What is fascinating is that he believes in this thing, which can not exist, yet he does not believe in the existence of the geometric "point", which is perfectly valid. And that was the original topic of this thread, brought over from the Philosophy section to see what the mathematicians say on the issue. If you look at the first page, they seem to think that the point exists, but the infinitesimal does not. PJ has side-stepped this refutation and is looking for any tidbit here (or on wiki) which will allow him to buttress his shaky faith.

So you are saying that the infintesimal represents the "quickness" they resolve to zero?

Basically, yeah. Any finite hyperreal number can be thought of as consisting of a "standard part" (which is a regular real number) plus an infintesimal. A hyperreal number corresponds to the class of all sequences of real numbers which converge to the same (real) limit in the same way. The standard part gives the limit, and the infintesimal part tells you about the asymptotic behavior of the sequence. The infintesimal encodes more than just the rate. For example, the sequences {1, 1/2, 1/4, 1/8, ...} and {2, 1, 1/2, 1/4, ...} both have limit 0, but different infintesimals. Even though they both decay at the same *rate*, the former is always exactly twice the latter. So, the ratio of the latter to the former would be 2.


Yet are not limits supposed to represent the constant movement towards, but never the resolution to, the limit so described?

No, the fact of taking a limit doesn't imply that the sequence in question doesn't reach the limit in a finite time. Consider the limit of the sequence {1, 1/2, 0, 0, 0, 0,...}. Clearly, the limit is attained in a finite number of steps. The more interesting limits are the ones that take more than a finite number of steps, but there's nothing in the definition of a limit that requires that.


Therefore the answer would never resolve to zero, even over an infinite process, but would rightfully be the infintesimal the process is so described as?

No, it gets to zero after an infinite numbers of steps. The infintesimal part just tells us how it got there.


Also, would we not be able to bypass the aforementioned issues regarding multiplication by zero and multiplication of zero by infinity?

Kind of. The whole point of working with infintesimals is that you can do arithmetic that you can't do with the regular zero and infinity. In particular, you can divide infintesimals with eachother and you can multiply them by infinite numbers. However, this doesn't actually get you anything new: the product of the hypereal number 0 (i.e., standard and infintesimal parts = 0, which is to say the sequence {0,0,0,...}) and its reciprocal (standard part = 00 and infintesimal part = 0, or {oo,oo,oo,...}) is still undefined. Likewise for 0/0. What you CAN do is consider similar operations for non-zero infintesimals. For example, you can divide a non-zero infintesimal by itself, and you'll get 1: {1,1/2,1/4,...}*{1,2,4,...} = {1,1,1,...}.

Let me emphasize, however, that this is simply shorthand for results that hold in regular real analysis. When we say that 0*oo is undefined, we don't mean that you can't take the limit of the product of a sequence approaching zero and a sequence approaching 00. What we mean is that the answer depends on the exact behavior of the sequences. In the above example, we get 1. But consider {1,1/2,1/4,...}*{1,4,16,...} = {1,2,4,...}, in which case the answer is infinity. You can also come up with examples where you'll get zero. It depends on which sequence converges to its limit faster. A hyperreal number, then, tells you both the limit of the sequence and its convergence properties, and so provides a convenient shorthand for these sorts of comparisons.

1. The aforementioned infinitesimal * infinity = finite value.

As I just mentioned, the result can also be infinite or infintesimal, depending on which infintesimal and which infinity you're talking about.


Aka: All finite values can be represented through an infinite series of infinitely small parts.

Not only that, there are an infinite number of ways of constructing said infinite series!


2. That infinitesimals are in fact, lower than any finite, non-zero, positive real-number.

Yes, but they aren't on the real line. Except for 0.


Aka: They are indivisble and can only be approached through an infinite process.

I don't know what you mean by indivisible; you can certainly divide them by other hyperreals. One doesn't "approach" a hyperreal; a hyperreal tells us how fast one is approaching a standard real.

But that is not what PJ claims. This debate, about whether or not a point has zero size, stems from a disagreement PJ and I have been having regarding the nature of his concept of an infinitesimal. He claims that there is a non-zero *fixed* value of which an infinite number can exist within a finite segment.
Such a partition of a closed subset of \mathbb{R} ^n doesn't exist. Also, infinitesimals aren't points - they have "infinitesimal measure", dimension, orientation, etc. They're probably best thought of as place-holders in differential equations that don't correspond to anything measurable in the physical world (your ruler will never tell you the length of anything is "infinitesimal"), and always get integrated away in the end.
This confusion arises when PJ imagines dividing that segment into smaller bits. Since he can imagine doing this "forever", with the bits getting smaller and smaller, he imagines that there is an infinite number of these small bits in a line segment, let's say between 0 and 1. My point is that this process may be eternal, but there exists a limit at 0 which prevents you from ever reaching a small number, of which there will be an infinite number of them in that segment of length 1.
I think you should decide whether it is infinitesimals or points you wish to talk about. If I had to venture a bold guess, I'd say it was impossible to split, say, an interval of the real number line into its constituent points by this infinite division into segments. The reals are uncountably infinite.
PJ feels that there must be such a number and has construed it as: .000...1 Which I have pointed out to him as being an infinite construct somehow bounded on both sides, which is impossible. He has lessened his usage of this construct, but still tries to concoct it by other means.
\lim_{n \rightarrow \infty} 10^{-n} exists and equals zero. I don't see how else to interpret ".000...1".
What is fascinating is that he believes in this thing, which can not exist, yet he does not believe in the existence of the geometric "point", which is perfectly valid. And that was the original topic of this thread, brought over from the Philosophy section to see what the mathematicians say on the issue. If you look at the first page, they seem to think that the point exists, but the infinitesimal does not. PJ has side-stepped this refutation and is looking for any tidbit here (or on wiki) which will allow him to buttress his shaky faith.
I'm not sure I see the point of this discussion. All I can say is that measure theory doesn't lead to contradictions and, on a more personal note, is generally compatible with my intuitive notion of "size" in the physical world. There's no way you can make objective claims about what the measure of a point "really" is or "should" be in mathematics.

x/oo = infinitesimal (infinity borrowing from the other poster who introduced that) where x is any non-zero, non-infinitesimal, real number or infinite number.

No, if x is an infinite (hyperreal) number, then the ratio could be infintesimal, finite or infinite. It depends on how fast the numerator and denominator go to infinity. For example, {1,2,3,...}/{1,2,3,...} = {1,1,1,...}, which is finite.

If x is a finite hyperreal, then, yes, division by an infinite hyperreal results in an infintesimal. For example, {1,1,1,...}/{1,2,4,...} = {1,1/2,1/4,...}. Notice that this infintesimal can be 0, for example if by oo you mean the reciprocal of 0 (i.e. the hyperreal corresponding to {oo,oo,oo,...}). Certainly, {1,1,1,...}/{oo,oo,oo,...} = {0,0,0,...}, which corresponds to the hypereal number 0.


The reason for the above being your mentioned "infinitesimal * infinity = any value" whereas if x/oo = 0, 0 * infinity = undefined or 0. Moreover, if it is 0 the process would be irreversible and therefore invalid, as all processes the quotient to any process of division, multiplied by the number one is dividing by, equals the number divided. So for instance, 1 / 2 = .5 * 2 = 1.

A process doesn't have to be reversible to be valid. Dividing by infinity is not reversible, but it's still well-defined. You get 0, and lose all the information about what you started with. This is no less true with hyperreal numbers. What you're interested in, however, is not so much division by oo, but an infinite sequence of divisions by some finite number. Each step, then, is reversible. In terms of hyperreals, this is written as division by an infinite number, but it's important to remember that there are multiple, distinct infinite numbers in the hyperreals (just as there are distinct infintesimals). It's only when you divide by an infinite hyperreal with a non-zero non-standard part that you get a non-zero result (i.e., the hyperreal corresponding to {1,2,3,...}). It's these "infinities" that correspond to your reversible processes of division, but the plain old oo is still included, and still produces 0 when you divide by it. And 0*oo is still undefined.

\lim_{n \rightarrow \infty} 10^{-n} exists and equals zero. I don't see how else to interpret ".000...1"

That was also my conclusion. The construct .000...1 == .000... If there is an infinite number of zeros, you never get to place the '1', so the result is zero.

Quadraphonics:

Basically, yeah. Any finite hyperreal number can be thought of as consisting of a "standard part" (which is a regular real number) plus an infintesimal. A hyperreal number corresponds to the class of all sequences of real numbers which converge to the same (real) limit in the same way. The standard part gives the limit, and the infintesimal part tells you about the asymptotic behavior of the sequence. The infintesimal encodes more than just the rate. For example, the sequences {1, 1/2, 1/4, 1/8, ...} and {2, 1, 1/2, 1/4, ...} both have limit 0, but different infintesimals. Even though they both decay at the same *rate*, the former is always exactly twice the latter. So, the ratio of the latter to the former would be 2.

Your explanation was clear and this seems like both a useful conception and one which has no flaws.

No, the fact of taking a limit doesn't imply that the sequence in question doesn't reach the limit in a finite time. Consider the limit of the sequence {1, 1/2, 0, 0, 0, 0,...}. Clearly, the limit is attained in a finite number of steps. The more interesting limits are the ones that take more than a finite number of steps, but there's nothing in the definition of a limit that requires that.

Okay. Let's have these finite limits. What about the non-finite limits? Would these resolve to zero even in infinite time? This would seem not to be the case if we can speak of infinitesimals lower than any non-zero, positive, real number. For if we can speak of these things, we can use them to reconstruct the aforementioned process in reverse and give a sensible answer, whereas we cannot do the same for resolution to zero.

In the process of infinite division which we have spoken of, to resolve it to zero would make the process irreversible. The dreaded 0 * infinity = ? or 0 (according to myself and from what I gather, probability theory). If an infinitesimal, specifically one that relates back to the ratios we just went over, we'd have the exact value of the process when multiplied back by the answer. Ergo, it usage is not only more useful, but "more correct" than zero. For the zero would not fulfill the demands of a quotient.

No, it gets to zero after an infinite numbers of steps. The infintesimal part just tells us how it got there.

See above.

Kind of. The whole point of working with infintesimals is that you can do arithmetic that you can't do with the regular zero and infinity. In particular, you can divide infintesimals with eachother and you can multiply them by infinite numbers. However, this doesn't actually get you anything new: the product of the hypereal number 0 (i.e., standard and infintesimal parts = 0, which is to say the sequence {0,0,0,...}) and its reciprocal (standard part = 00 and infintesimal part = 0, or {oo,oo,oo,...}) is still undefined. Likewise for 0/0. What you CAN do is consider similar operations for non-zero infintesimals. For example, you can divide a non-zero infintesimal by itself, and you'll get 1: {1,1/2,1/4,...}*{1,2,4,...} = {1,1,1,...}.

So yes: Infinitesimals can do what zero cannot?

Let me emphasize, however, that this is simply shorthand for results that hold in regular real analysis. When we say that 0*oo is undefined, we don't mean that you can't take the limit of the product of a sequence approaching zero and a sequence approaching 00. What we mean is that the answer depends on the exact behavior of the sequences. In the above example, we get 1. But consider {1,1/2,1/4,...}*{1,4,16,...} = {1,2,4,...}, in which case the answer is infinity. You can also come up with examples where you'll get zero. It depends on which sequence converges to its limit faster. A hyperreal number, then, tells you both the limit of the sequence and its convergence properties, and so provides a convenient shorthand for these sorts of comparisons.

Might give an example where any infinitesimal * oo = 0?

As I just mentioned, the result can also be infinite or infintesimal, depending on which infintesimal and which infinity you're talking about.

Granted, although I'll be looking forward to the infinitesimal * oo = 0 proof.

Not only that, there are an infinite number of ways of constructing said infinite series!

Indeedio.

Yes, but they aren't on the real line. Except for 0.

Yes. They are hyperreal. This much is certain. And zero also has the quality of being lower than any other infinitesimal.

I don't know what you mean by indivisible; you can certainly divide them by other hyperreals. One doesn't "approach" a hyperreal; a hyperreal tells us how fast one is approaching a standard real.

Well in absolute value, certainly an infinitesimal cannot be further reduced, yes? That is to say, though it will tell you how "fast" it is getting there, all infinitesimals in absolute value will be equal to one another and closest to zero without being such that can be, yes?

Okay. Let's have these finite limits. What about the non-finite limits? Would these resolve to zero even in infinite time?

Yes. The limit of {1,1/2,1/4,...} is the real number zero. The reals are closed: you cannot get a hyperreal number as the limit of a sequence of real numbers.


This would seem not to be the case if we can speak of infinitesimals lower than any non-zero, positive, real number. For if we can speak of these things, we can use them to reconstruct the aforementioned process in reverse and give a sensible answer, whereas we cannot do the same for resolution to zero.

All infintesimals are "resolving to zero" by definition. The extra information they contain about how they get to zero, however, lets one consider inverting them. That is, there is an infinite hyperreal defined by the reciprocal of every infintesimal, and their product is 1 (unless you're talking about 0 and oo, which is still undefined).


In the process of infinite division which we have spoken of, to resolve it to zero would make the process irreversible. The dreaded 0 * infinity = ? or 0 (according to myself and from what I gather, probability theory). If an infinitesimal, specifically one that relates back to the ratios we just went over, we'd have the exact value of the process when multiplied back by the answer. Ergo, it usage is not only more useful, but "more correct" than zero. For the zero would not fulfill the demands of a quotient.

It's not that the limit of the process is non-zero. It is zero. However, the process can be encapsulated by a hyperreal number, which allows you to define operations such as infintesimal*infinite = finite. It's not that 0*oo = undefined prevents one from reversing a recursive division. The fact that it's undefined is just a reminder that we need to know how fast "0" went to zero, and how fast "oo" went to infinity. Hyperreal numbers contain this information, which then lets you write the answer using simple arithmetic. But it's certainly true in standard real analysis that {1,1/2,1/4,...}*{1,2,4,...} = {1,1,1,...}. The fact that the limit is zero doesn't imply anything about how the sequence got there, and so has no bearing on "reversibility." Irreversibility would correspond to a sequence that hits 0 in some finite number of steps.


Might give an example where any infinitesimal * oo = 0?

When you say "oo," do you mean the hyperreal number corresponding to {oo,oo,oo,...} or just any infinite hyperreal number (i.e., {1,2,3,...}). If you mean the former, then no, you can't multiply that by anything and get zero. If you mean the latter, it's easy:

{1,1/4,1/16,...}*{1,2,4,...} = {1,1/2,1/4,...}

Or, better yet, {0,0,0,...}*{1,2,4,...} = {0,0,0,...}. Keep in mind that 0 IS an infintesimal.


Well in absolute value, certainly an infinitesimal cannot be further reduced, yes? That is to say, though it will tell you how "fast" it is getting there, all infinitesimals in absolute value will be equal to one another and closest to zero without being such that can be, yes?

No, you can order the infintesimals. That's the whole idea. It wouldn't make sense to talk about there being multiple non-zero infintesimals if they were all indistinguishable. For example, the infintesimal corresponding to {1,1/2,1/4,...} is smaller than the one corresponding to {1,1/4,1/9,...}. Which is to say that if you divide the latter by the former, you get an infinite hyperreal. Which is to say that some infintesimals are infinitely larger than others (although the distance between them is infintesimal). Which means that, indeed, you can further reduce an infintesimal. In fact, you can keep on breaking them into smaller pieces *for ever*.

Swivel:

You can't divide by infinity.

And please tell me how an infinite number of non-zero "bits" can exist within a finite space. You are wrong on this. It would suppose a finite number, which can be multiplied by infinity, and NOT get infinity. Never happen.

Evidently one can, according to Quadraphonics and myself.

But let's see here, any finite, positive, non-zero real number can be divided over an infinite process.

The answer to this process is an infinitesimal.

An infinitesimal, multiplied by infinity, can give any value.

Ergo, you are wrong that any finite value cannot have infinite parts.

And thanks for admitting one of your mistakes. It warms my heart to see you making some progress here. Since you often used as a "proof" your infinite distance scenario, this admission should allow you to discard the rest of your conclusions.

My proof has been revised, yes. HOwever, the rest of my conclusions are seemingly quite right.

Regarding your other point:

All infinitesimals have an absolute value (regardless of the ratio they represent) that is non-zero but beneath any finite value. Accordingly, it is exactly what you have argued against: Further indivisble (in absolute value) and essentially .000...1 (or a value of infinite division).

przyk:

As I've been arguing, the notion of a point which is held to be as it is orthodoxically considered is incoherent. Consider that any finite real number is infinitesimally greater or lesser than the numbers bounding it on either side. Accordingly, to represent it on a grid and ascribe its point a zero-size, would not reflect that it's value is defined down to the infinitesimal.

I'm also debating this with someone else, so please see my responses to him for the rest of my argument thus far.

Quadraphonics:

No, if x is an infinite (hyperreal) number, then the ratio could be infintesimal, finite or infinite. It depends on how fast the numerator and denominator go to infinity. For example, {1,2,3,...}/{1,2,3,...} = {1,1,1,...}, which is finite.

A question, though: Would not this only work if we are dealing with an infinte process/infinite process, rather than the value of infinity/value of infinity? For what we are getting there clearly is a 1:1 ratio, but this is not the process of division.

If x is a finite hyperreal, then, yes, division by an infinite hyperreal results in an infintesimal. For example, {1,1,1,...}/{1,2,4,...} = {1,1/2,1/4,...}. Notice that this infintesimal can be 0, for example if by oo you mean the reciprocal of 0 (i.e. the hyperreal corresponding to {oo,oo,oo,...}). Certainly, {1,1,1,...}/{oo,oo,oo,...} = {0,0,0,...}, which corresponds to the hypereal number 0.

I am afraid I don't grasp why {1,1,1,...}/{oo,oo,oo,...} = {0,0,0,...}? Would not every 1, divided by by the corresponding oo, always be an infinitesimal? Why would it, taken together, = 0,0,0?

A process doesn't have to be reversible to be valid. Dividing by infinity is not reversible, but it's still well-defined. You get 0, and lose all the information about what you started with. This is no less true with hyperreal numbers. What you're interested in, however, is not so much division by oo, but an infinite sequence of divisions by some finite number. Each step, then, is reversible. In terms of hyperreals, this is written as division by an infinite number, but it's important to remember that there are multiple, distinct infinite numbers in the hyperreals (just as there are distinct infintesimals). It's only when you divide by an infinite hyperreal with a non-zero non-standard part that you get a non-zero result (i.e., the hyperreal corresponding to {1,2,3,...}). It's these "infinities" that correspond to your reversible processes of division, but the plain old oo is still included, and still produces 0 when you divide by it. And 0*oo is still undefined.

I fail to see how this is true? For if we are dealing with the value of a division by infinity, the reversibility would be found in the fact that to check whether or not this is true, we'd have to multiply it by the number it was divided by and get back to it. If it isn't, then clearly this is wrong, no? But if it is an infinitesimal such as the infinitesimals we get from the other process, we can indeed get the number back, or if not, we can ascribe it on the equality of the absolute values of the hyperreal infinitesimals and infinite numbers, no?

Also, I am not trying to refute the notion that any infinite process would have to be reversible within "another infinite span of time". Only that the value can be dealt with regardless. To clarify, that is.

Consider that any finite real number is infinitesimally greater or lesser than the numbers bounding it on either side.

Though I lack the expertise that others clearly demonstrate in this thread, I debate this point.

A finite real number is finitely separated by the numbers bounding it on either side if you consider the precision of the number you're talking about, and the precision of the numbers that bound it. Maybe it's accuracy, I get them backwards all the time.

Regardless, the numbers on either side do not exist unless you choose them, then it can be said whether or not they bound the number in question. In fact, based on the number in question, any finite real number, it's easy to choose numbers that bound it to the precision required by the purpose being undertaken.

I think what you're saying {2 * (infinitesimal) + x} != x ... (1 inf for dist to number below and 1 for above)?

Seeing as how that if quadraphonics is correct in saying zero is an infinitesimal, at least in that case you're clearly wrong... if indeed that's what you're saying, which it must be as far as I can tell.

Quadraphonics, is zero the only infinitesimal in the real set?

But let's see here, any finite, positive, non-zero real number can be divided over an infinite process.

The answer to this process is an infinitesimal.

No, the "answer" is zero. You cannot get to an infintesimal out of arithmetic operations on real numbers, even if you are allowed an infinite number of operations. This is part of the definition of real numbers. Rather, the infintesimal is a description of how the process gets to zero. Any method of partitioning a line segment into infinitely many disjoint pieces requires that each piece has size zero. The infintesimal simply keeps track of how "big" an infinity of pieces you end up with. It is emphatically incorrect to imagine the real line as being composed some infinite number of infintesimals. Infintesimals are descriptions of the division process itself, not its result. They do not live in the same universe as the real line.


Further indivisble (in absolute value)

No, as I've just shown, you can keep on dividing infintesimals for ever. Just as there is no smallest non-zero real number, there is no smallest non-zero infintesimal. I can always construct a sequence that converges to zero faster than any example you might care to consider. Which is to say that one cannot hope to arrive at the "smallest" element of a line through a countable process of division. No matter how many divisions you do, there will still be an uncountable quantity of real numbers in your line segment.

It bears repeating that hyperreals are nothing more than a shorthand for various computations that are perfectly well defined in standard real analysis. They are constructed such that they cannot disagree with any of the results of real analysis. They are used only for convenience. I would recommend forgetting about them until you've already become comfortable with standard analysis, as they seem to be confusing you. All of your questions can be phrased and answered in terms of standard analysis (and have been in this thread), and non-standard analysis will always agree with these answers. If it seems otherwise, it's because you're misunderstanding what hyperreal numbers represent.

Quadraphonics, is zero the only infinitesimal in the real set?

Yes.

A question, though: Would not this only work if we are dealing with an infinte process/infinite process, rather than the value of infinity/value of infinity?

Yes. The hyperreals correspond to infinite processes. The "value of infinity," in terms of hyperreals, corresponds to the sequence {oo,oo,oo,...}. Likewise, 0 corresponds to the sequence {0,0,0,...}. You cannot divide this infinity by itself (or multiply it by 0). Just as in the case of real numbers, these operations are undefined. However, there are other infinities in the hyperreals, such as {1,2,3,...}, for which these operations are well-defined. Hyperreals don't avoid the lack of definition for things like 0*oo; rather, they introduce a bunch of *new* zeros and infinities for which these operations can be defined. But it is still the case that the usual 0/0, 0*oo, oo/oo, etc. are undefined.


I am afraid I don't grasp why {1,1,1,...}/{oo,oo,oo,...} = {0,0,0,...}? Would not every 1, divided by by the corresponding oo, always be an infinitesimal?

Yes, and that infintesimal is 0. To get a non-zero infintesimal, you'd have to use an infinity of the form {1,2,4,...}. I.e., {1,1,1,...}/{1,2,4,...} = {1,1/2,1/4,...}, which is a non-zero infintesimal.


I fail to see how this is true? For if we are dealing with the value of a division by infinity, the reversibility would be found in the fact that to check whether or not this is true, we'd have to multiply it by the number it was divided by and get back to it. If it isn't, then clearly this is wrong, no? But if it is an infinitesimal such as the infinitesimals we get from the other process, we can indeed get the number back, or if not, we can ascribe it on the equality of the absolute values of the hyperreal infinitesimals and infinite numbers, no?

Well, as the examples above indicate, it depends on which infinity you're dividing by. Dividing by the cannonical oo results in exactly 0, and so is not reversible. So we would not expect 0*oo to be defined. Doing an infinite series of divisions by some finite real number corresponds to a different infinity, and these are reversible. That is to say, {1,1/2,1/4,...}*{1,2,4,...} = {1,1,1,...}, whereas {0,0,0,...}*{oo,oo,oo,...} = undefined. Just as the non-zero infintesimals have slightly different properties than zero, the various infinite hyperreal numbers other than the cannonical oo have slightly different properties. The regular old 0 and oo are still present, however, and their properties don't change (which is required for non-standard analysis to agree with standard analysis).

PJ, you are really stretching in this thread. Grasping at hyperreals to prove your point about reals is not a good sign. Evidently what you are doing is agreeing with anything in quadraphonics excellent posts that give your bizarre theory some breathing room, and rejecting those things that shoot it down. This is how pseudosciences are built.

http://mathforum.org/library/drmath/view/62486.html

If you look at this:

http://www.math.utah.edu/~pa/math/0by0.html

You can see how infinity works the same way.

Since 2/oo = 3/oo But 2 != 3, we have a problem. Cantor confused this by showing that some infinities are bigger than others, and since then, mathematicians have been finding more ways to treat infinite sets as if they are numbers. You simply can't. Not in the way you are doing it.

I admire the way your mathematical vocabulary has expanded in your quest for validation of an awful set of assumptions, but you are working the same way any other psuedoscientist would work. You are starting with your conclusion and working backwards towards any observations that will fit.

I suggest you also try reading some things that you will disagree with, instead of googling for validation of an incorrect theory. Take in both sides if you really want to stand a chance of finding the truth of things:

http://www.marypat.org/stuff/nylife/010524.html

Here's a quintessential limit: What is the limit of 1/x as x goes to
infinity? Well, lots of people say that any finite number divided by
infinity is 0, so the answer is 0. And they'd be =wrong=. Because you
can't divide by infinity, because infinity is not a number. Are we
straight on that? For then that means infinity is the inverse of 0, and
we can't be having that now, can we? Let's keep our fields in good
condition.

Wesmorris:

So your critique is essentially one of pragmatism, in regards to your critique based on "exist until you choose them"?

I think what you're saying {2 * (infinitesimal) + x} != x ... (1 inf for dist to number below and 1 for above)?

Seeing as how that if quadraphonics is correct in saying zero is an infinitesimal, at least in that case you're clearly wrong... if indeed that's what you're saying, which it must be as far as I can tell.

I am afraid I don't grasp this point in how it:

1. Relates to mine.

2. What the equation refers to?

Quadraphonics:

No, the "answer" is zero. You cannot get to an infintesimal out of arithmetic operations on real numbers, even if you are allowed an infinite number of operations. This is part of the definition of real numbers. Rather, the infintesimal is a description of how the process gets to zero. Any method of partitioning a line segment into infinitely many disjoint pieces requires that each piece has size zero. The infintesimal simply keeps track of how "big" an infinity of pieces you end up with. It is emphatically incorrect to imagine the real line as being composed some infinite number of infintesimals. Infintesimals are descriptions of the division process itself, not its result. They do not live in the same universe as the real line.

However can this be, though? If you have an infinity of zero-sized parts, you resolve back to that undefined "zero * infinity", which would not satisfy the division of the line to begin with, no? An "infinity of zero-sized segments" is itself a contradiction of terms, even.

Also, doesn't your conception of the infinitesimals contradict the notion that infinitesimals are "hyperreal numbers which aren't zero but which are always less than any finite numbers"? This does not seem to imply they are greater than zero nor resolvable purely to it, no?

Quadraphonics:

Yes. The hyperreals correspond to infinite processes. The "value of infinity," in terms of hyperreals, corresponds to the sequence {oo,oo,oo,...}. Likewise, 0 corresponds to the sequence {0,0,0,...}. You cannot divide this infinity by itself (or multiply it by 0). Just as in the case of real numbers, these operations are undefined. However, there are other infinities in the hyperreals, such as {1,2,3,...}, for which these operations are well-defined. Hyperreals don't avoid the lack of definition for things like 0*oo; rather, they introduce a bunch of *new* zeros and infinities for which these operations can be defined. But it is still the case that the usual 0/0, 0*oo, oo/oo, etc. are undefined.

So essentialyl, they provide different answers to different questions?

Yes, and that infintesimal is 0. To get a non-zero infintesimal, you'd have to use an infinity of the form {1,2,4,...}. I.e., {1,1,1,...}/{1,2,4,...} = {1,1/2,1/4,...}, which is a non-zero infintesimal.

Although I have obvious qualms related back to it, I can see where this works given my answer being wrong (which I am not ready to concede).

Thanks for the clarification.

Well, as the examples above indicate, it depends on which infinity you're dividing by. Dividing by the cannonical oo results in exactly 0, and so is not reversible. So we would not expect 0*oo to be defined. Doing an infinite series of divisions by some finite real number corresponds to a different infinity, and these are reversible. That is to say, {1,1/2,1/4,...}*{1,2,4,...} = {1,1,1,...}, whereas {0,0,0,...}*{oo,oo,oo,...} = undefined. Just as the non-zero infintesimals have slightly different properties than zero, the various infinite hyperreal numbers other than the cannonical oo have slightly different properties. The regular old 0 and oo are still present, however, and their properties don't change (which is required for non-standard analysis to agree with standard analysis).

Let me ask this question in regards to the value of X/oo.

From what I gather, the answer X/oo = 0 is given because it would be the natural reciprocal to X/0 = oo, yes?

Swivel:

As your critiques are offered in the interest of rationality, I do not take offense that you are attempting to label me a pseudo-scientist. But know that I have indeed read things that have disagreed with my premise.

However, is not your mathforums webpage explicitly refuted by the proofs Cantor gave of "greater infinities"? Even if you evidently agree with him, it would seem his proofs would deal with all the objections given within the two things you linked me to.

However, is not your mathforums webpage explicitly refuted by the proofs Cantor gave of "greater infinities"?

In what way? The mathforums page talks about things like 1/infinity. Cantor proved that there are different "sizes" of infinitely large sets. The first has nothing to do with the latter. Since Cantor, the concept of cardinal numbers has been developed. The cardinal numbers are not the reals; they are a beast of a completely different color. For example, subtraction and division cannot be defined for the cardinals.

Wesmorris:

So your critique is essentially one of pragmatism, in regards to your critique based on "exist until you choose them"?

In the real set, zero is the only infinitessimal. As such, any other "bounding numbers" or however you phrased it originally, must be defined in order to discuss it. There is no "infinitesimally greater or lesser than the numbers bounding it on either side" besides the x+the infinitesimal (which is zero) and x-the infinitesimal (which is zero), leaving you with simply X, as I discussed above.

I am afraid I don't grasp this point in how it:

1. Relates to mine.

Hopefully I've clarified.

2. What the equation refers to?

The point infinitessimally above x would be (x+infinitesimal). The point below would be (x-infinitesimal). Thus the magnitude of the range would be [2 * (infinitesimal)]. You said in the statement I courtiqued that "any finite real number is infinitesimally greater or lesser than the numbers bounding it on either side. ". Translating into a simple equation, you're saying that:

{[2 * (infinitesimal)] + x} != x

(the range of the numbers that bound x is greater than x)

Whereas since 0 is the only infinitesimal in the real set, you are clearly mistaken, since:

2*0 + x = x

x = x

As I've been arguing, the notion of a point which is held to be as it is orthodoxically considered is incoherent.
A point is an element of \mathbb{R}^n. The Lebesgue measure of a point is zero. Please show how this leads to a logical contradiction.
Consider that any finite real number is infinitesimally greater or lesser than the numbers bounding it on either side.
Real numbers aren't "bounded" on either side. Pick any two real numbers x and y, and I will always be able to find one in between, for example by taking \frac {x + y} 2. It really is that simple.

Please don't try to bring infinitesimals into this. The whole point of a measure is to assign non-negative real numbers to describe the "size" of a set, which is what makes the concept useful in science and engineering. A length of "dx" is physically meaningless.
Accordingly, to represent it on a grid and ascribe its point a zero-size, would not reflect that it's value is defined down to the infinitesimal.
What does this mean? Go on - translate this into a statement rigourous enough to satisfy a mathematician.
I'm also debating this with someone else, so please see my responses to him for the rest of my argument thus far.
I've been following the thread.

Swivel:

As your critiques are offered in the interest of rationality, I do not take offense that you are attempting to label me a pseudo-scientist. But know that I have indeed read things that have disagreed with my premise.

However, is not your mathforums webpage explicitly refuted by the proofs Cantor gave of "greater infinities"? Even if you evidently agree with him, it would seem his proofs would deal with all the objections given within the two things you linked me to.

You are completely clueless when it comes to these issues, as a glance back at the two other threads just reminded me. You have a preconceived idea, and you are desperately trying to find validation for it.

Cantor's sets are not numbers, they are sets. You are using sets and hyperreals instead of sticking with real numbers, which is where your problem lies. Stop running away from your problems.


Here is something you could learn from sets:

Imagine two sets. One set we will call SIZE. The other set, we will call NUMBER. Here is what each set contains:

SIZE is made up of the size of our units of measurement as we keep dividing them in half to find the "smallest" one. It goes [1, .5, .25, .125, ...] The other set, NUMBER contains the number of each unit of measurement that we have in our unit line. It goes [1, 2, 4, 8, ...].

Now, your biggest confusion comes from the idea that the set SIZE has a countless number of members, which you confuse with the presence of a size so small that an infinite number can exist in that unit line segment. Wrong, wrong, wrong.

The set NUMBER is the exact same size as the set SIZE. For each member of SIZE, there is a corresponding member of NUMBER. Both sets go on forever, but just as SIZE will never reach Zero, NUMBER will never reach infinity. There will always be a well-defined real value in each set which correspond with each other.

I have explained this to you before without the reference to sets, and keep reminding you that the *Eternal* process of division does not mean an infinitesimal is created, nor does it mean that the count (the members of NUMBER) ever reaches infinity.


You need to get over this desire of yours so that we can move on to the zero-length of a point, which is what this thread is all about. You are wrong on that count as well, and it would please me to see you come about on both issues.

So essentialyl, they provide different answers to different questions?

No, they provide the same answers to the same questions. Hyperreals are just a shorthand for classes of sequences of real numbers, and the results of the various operations on hyperreals are just shorthand for the results of the corresponding operations on sequences of real numbers. The hyperreals are defined in such a way that they can never disagree with real analysis, nor can they be used to prove anything beyond the results of real analysis.


Let me ask this question in regards to the value of X/oo.

From what I gather, the answer X/oo = 0 is given because it would be the natural reciprocal to X/0 = oo, yes?

That particular reciprocal relationship holds, yes, but that's not how those expressions are defined. In both cases, you start with some finite real number X and then take the limit as the denominator approaches oo (or 0, in the latter case). You then find that said limit is 0 (or oo, in the latter case), and so can then define the expression x/oo = 0 (and vice-versa). 0*oo remains undefined.

However can this be, though? If you have an infinity of zero-sized parts, you resolve back to that undefined "zero * infinity", which would not satisfy the division of the line to begin with, no?

No. It's not a problem that the arithmetic operation 0*oo is undefined, because the standard analysis doesn't require us to perform that operation. To be explicit, consider the segment of the real line [0,1]. We start with a single segment with size 1. Now, we split it in half, giving us 2 segments of length 1/2. Then, we split those in half, giving 4 segments of length 1/4, and so on. Thus, we have two sequences: sequence a(n) gives the number of segments at step n, and sequence b(n) gives the length of each segment at time n. I.e.:

a(n) = {1,2,4,8,...}
b(n) = {1,1/2,1/4,1/8,...}

Clearly, the limit of a(n) is oo and the limit of b(n) is 0, so we're going to end up with infinitely many pieces of length 0. Now, reversing the process corresponds to multiplying the two sequences together, which results in the sequence {1,1,1,...}. The limit of this sequence is clearly 1, which is to say that we aren't losing any pieces of the original line segment.

The point is that you have to do the limit *after* multiplying the sequences together. The statement "0*oo" refers to the limit of the product ((NOT the product of the limits!) of two sequences, the first of which has limit 0 and the second of which has limit oo. When we say that the expression 0*oo is undefined, we don't mean that you can't work out the limit of the product of such sequences. We mean that said limit could take on any number of values depending on how quickly the respective sequences approach their limits. This isn't a problem, it just means that you have to sit down and examine the details of the sequences to arrive at an answer (as we did above).

Conversely, we can define expressions such as x/oo = 0 because they do not depend on how the denominator got to its limit. The limit of a finite x divided by any sequence which approaches oo will be 0, regardless of how said sequence gets to oo. Expressions like 0*oo, on the other hand, depend on the relative rates of convergence of the two terms. If the "0" sequence is faster, you'll get 0, and vice versa. If the two sequences have the same rates of convergence, then you'll get a constant (as we did in the above example). Hyperreal numbers simply provide a shorthand for these considerations by embedding information about how quickly the sequences approach their limits. But there's nothing new here. When you say "the product of some particular infintesimal and some particular infinite hyperreal is a finite hyperreal," it's just shorthand for the statement "the limit of the product of a particular real sequence with limit 0 and another particular real sequence with limit oo is a constant."


An "infinity of zero-sized segments" is itself a contradiction of terms, even.


How so?


Also, doesn't your conception of the infinitesimals contradict the notion that infinitesimals are "hyperreal numbers which aren't zero but which are always less than any finite numbers"? This does not seem to imply they are greater than zero nor resolvable purely to it, no?

I don't understand what you mean by this.