If there is no BIGGEST number, is there a SMALLEST?

[C1ay]

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

:) 0 * anything is 0. All conversations of infinity aside, that much should be learned in elementary school.

[Boerseun]

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

0 x infinity will always be zero.

Some wouldn't want to define infinitesimals directly from infinity, as you did: 1/INF. In my view, if you don't define infinitesimals by way of infinity, you're not really defining infinitesimals. As far as I know, mathematics has NOT yet defined TRUE infinitesimals.

The smallest number above zero would have to be 1/infinity. It stands to reason.

Even though you may agree with

 

1 + x = 1 , x being an infinitesimal x =1/INF.

 

most mathematicians don't agree with that. The evidence: Mathematics still says that for any "infinitesimal" x: 1+x > 1. Yet, ironically, for an infinite X: X+1=1.

For any 1+x=1, x = zero. Not an infinitesimal, but ZERO. Otherwise it would've been 1+x>0. Not only does it stand to reason, it should be glaringly obvious.

[Guest loarevalo]

0 x infinity will always be zero.

 

The smallest number above zero would have to be 1/infinity. It stands to reason.

 

For any 1+x=1, x = zero. Not an infinitesimal, but ZERO. Otherwise it would've been 1+x>0. Not only does it stand to reason, it should be glaringly obvious.

 

Mathematics expressly say that 0*INF is indeterminate, like 0/0 is indeterminate - actually, both forms are equivalent, which should give you a clue.

 

You all seem to still miss the point:

If you agree that INF + 1 is NOT greater than (but equal to) INF,

why do you insist in that an INFinitesimal + 1 > 1 ?

 

It is in the "infinite" nature of Infinity that if you add 1 to it, that 1 will not enlarge Infinity - otherwise it wouldn't be infinite, right?

Likewise, it is by the "infinite" nature of an infinitesimal that if you add that infinitesimal to 1, the infinitesimal will not enlarge 1.

 

As a finite number adds nothing to infinity, so does an infinitesimal add nothing to a finite. To me, it's plain common sense. :)

[C1ay]

Your claims contradict this definition of Infinity. Time to test your math prowess while you show us proof of your claims loarevalo.

[Guest loarevalo]

Your claims contradict this definition of Infinity. Time to test your math prowess while you show us proof of your claims loarevalo.

Sorry, you may have to elaborate on that C1ay. I read the article on Mathworld about infinity and I can't see how anything I have said contradicts it. Actually, that was a pretty good concise article and I specially aplaud:

 

"Infinity is a very tricky concept to work with, as evidenced by some of the counterintuitive results that follow from Georg Cantor's treatment of infinite sets."

 

Still, I don't know what you mean that my claims contradict the definition of infinity. Did you read post #20 (about the *FACTS*)? if not, read it, specially the last fact about Calculus.

http://hypography.com/forums/showthread.php?t=3360&page=2&pp=10

 

The only definition of infinity I read in the Mathworld article is:

1/0 = INF and 1/INF = 0 (by way of limits), where did I say anything to the contrary? :)

[C1ay]

The only definition of infinity I read in the Mathworld article is:

1/0 = INF and 1/INF = 0 (by way of limits), where did I say anything to the contrary? :)

0 * INF isn't necesarily infinity, or zero, it could be a whole number too.

This contradicts the definition of infinity since you imply that infinity is some special number that can be multiplied with 0 and not equal 0.

[Guest loarevalo]

This contradicts the definition of infinity since you imply that infinity is some special number that can be multiplied with 0 and not equal 0.

C1ay agrees on this definition of infinity:

 

A=lim 1/x [as x->0] = INF

and

lim 1/x [as x->INF] = 0

 

Let's multiply INF * 0:

 

B = lim sqrt(x) [as x->0] = 0

("sqrt" means square root)

 

lim A*B = lim sqrt(x)*1/x [as x->0] = INF

so, 0*INF = INF

 

lim sin(x)/x [as x->0] = 1

so, 0*INF = 1

 

lim x^2 * 1/x [as x->0] = 0

so, 0*INF = 0 :)

 

If infinity is defined by limits, then is calculus fact that 0*INF is indeterminate. Where's the contradiction? Check your Calculus book.

 

I understand what you mean C1ay:

Infinity is like any number, incomplete in a way, so no matter how many times you add nothing, even if it's infinite times, you still get nothing.

 

Under this view, you assume "zero" to be the SMALLEST number, the Absolute Zero.

I do agree with you: Abs. Zero * INF = Abs. Zero.

We can say the same about the Absolute Infinity: Abs. INF * 0 = Abs. INF.

These operations on the absolutes, however, imply contradictions, and that is why the Absolute Infinity isn't defined in Set Theory - and why there shouldn't be an Absolute Zero.

[Guest loarevalo]

As a finite number adds nothing to infinity, so does an infinitesimal add nothing to a finite. To me, it's plain common sense. :rant:

 

What do you all think? Is that property of the infinitesimal at least expected, if not provable?

 

Why is it considered a paradox that

a quantity greater than an absolute nothing (but infinitesimal) + 1 is equal to 1,

and not that

a quantity less than an absolute everything (but infinite) + 1 is equal to the same infinite quantity?

 

Isn't it time to lay to rest that 300 year old paradox, which finally destroyed all confidence in the infinitesimal?

[Guest loarevalo]

So you are saying that it is OK that an infinitesimal is a number that is greater in absolute value than zero yet smaller than any positive real number, thus an infinitesimal x ≠ 0?

Does x/x = 1 because x > 0 or is it undefined because x = 0?

 

If we agree in the restricted definition of zero as "zero is the smallest number", then like you all say, an infinitesimal would be greater than zero, thus an infinitesimal x<>0.

 

Nonetheless, zero or the symbol "0" isn't defined so restrictedly. I think most people accept the definition of "0" to be x, the solution of the equation n + x = n, where n is a natural number (or any finite number).

 

Likewise, we could define a number "Ω" to be "Ω is the biggest number," then we would agree: an infinite number would be less than Ω, thus an infinite x <> Ω. But, we couldn't prove that Ω is the solution to X + n = X, n is a finite number, because ANY infinite would also be a solution. Rather than complicating so much, we often just say that the solution of X + n = X is simply INFINITY - the vague primitive idea of infinity. Obviously then, if INF is defined as the solution to that equation, we can see how every infinite number = INF, and Ω = INF, or rather INF is indeterminate. Set Theory establishes that such a "Ω" can NOT exist - as it should be obvious: there is no biggest last number.

 

Then. I only wished to apply this greater understanding of infinity, to infinitesimals and zero. Post #1 is a summary of that effort.

 

Thus, an "smallest" number doesn't exist, and if we restrictedly define zero as such number, then zero doesn't exist. But the symbol "0" is not so defined, but we have thought that this "0" would at the same time be the smallest number. As I discovered, by "0" we more accurately mean "infinitesimal," or 0 and infinitesimal are synonims - and thus 0 is indeterminate in that sense. And from that derives a landslide explanation of div-by-zero.

 

Is it more clear now? I'm just projecting the complexity of infinity unto the infinitesimal.

[Guest loarevalo]

This contradicts the definition of infinity since you imply that infinity is some special number that can be multiplied with 0 and not equal 0.

You are implying that 0 is a special number above all numbers, and that 0 is the absolute smallest number. What about the absolute biggest number? Anyhow, it is in the books that 0*INF is indeterminate, thus INF is as special as 0 :lol:

[Dundasbro]

I don't define 0 as much of a number, just the absence of one. When there is nothing, it's defined as 0.

 

And infinity is just the definition of a number that is ongoing, it can't end, it can't have a set amount and it can't be divided by zero. (Someone feel free to prove me wrong, I probebly am)

[infamous]

 

And infinity is just the definition of a number that is ongoing, it can't end, )

Actually, I personally believe that the concept of infinity can't be described as a number. We can theorize about the implications of the concept and we can also give it idenity by ascribing a sign to it, but, can anyone really express it in numerical form? I think not. Infinity may only exist in the mathematical imaginations we conjure up to express the unknowable extent of natures mysteries.

[Dundasbro]

Great point made, does anyone know of anything that can actually be defined as being infinate?

[Guest loarevalo]

Actually, I personally believe that the concept of infinity can't be described as a number. We can theorize about the implications of the concept and we can also give it idenity by ascribing a sign to it, but, can anyone really express it in numerical form? I think not. Infinity may only exist in the mathematical imaginations we conjure up to express the unknowable extent of natures mysteries.

It's true that we can't just ascribe a sign to infinity and *puff* magic, we have the number "infinity" - doing so, then we must define operations for this "infinity" (otherwise we couldn't use it) and there's where the inconsistencies begin. I think all/most of those inconsistencies derive from establishing such number as the biggest number. Set Theory clarified those paradoxes and sought to define infinity (or infinite numbers) as definite numbers ("aleph_one" is as much a number as 2 or 43).

See post #20 of this thread for more about actual infinities and Set Theory:

http://hypography.com/forums/showpost.php?p=51152&postcount=20

[Guest loarevalo]

And infinity is just the definition of a number that is ongoing, it can't end, it can't have a set amount and it can't be divided by zero. (Someone feel free to prove me wrong, I probebly am)

The potential infinity denoted ∞ fits what you said about "a number that is ongoing" - this very phrase indicates us the difficulty we have with things like infinity: Is an infinite number the same number always? or is it ongoing? That phrase seems similar to how Newton talked about infinitesimals as "fluxions." That way of thinking about infinity is better described as "potential" rather than "actual" - the Limit uses this concept of potential infinity.

 

Under the treatment that zero usually gets (that zero is the smallest number) no number can be divided by zero, or which is equivalent, any number multiplied by zero = zero. Except any such number as the biggest number, every number * 0 = 0. For the biggest number, these operations are indeterminate. Of course, neither the biggest number exist nor the smallest number, so there is no need to define operations with them.

[Guest loarevalo]

Great point made, does anyone know of anything that can actually be defined as being infinate?

To go straight to the point,

 

Actual infinite:

 

-The count of natural numbers. {1,2,3,...}

 

-The count of points in a continuum

(if you believe space to be a true continuum - so there is no indivisible particle - you believe in an actual infinity in physics, the number of positions or coordinates in space).

 

These two lines have equal number of points, though different measures:

_____________________

__________

Remember: INF * 2 = INF

[Guadalupe]

Great point made, does anyone know of anything that can actually be defined as being infinate?

 

 

How about outside our universe? :lol:

 

 

:hihi: