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A paradox inside Newtonian world 
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#181
Nov2206, 11:46 AM

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#182
Nov2206, 11:56 AM

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Limit for what? For the "leftmost" ball speed after a second? For the force between the two "leftmost" balls?
Nothing like that exists. 


#183
Nov2206, 01:02 PM

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#184
Nov2206, 02:16 PM

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#185
Nov2206, 04:02 PM

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#186
Nov2206, 05:13 PM

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This discussion seems to be an ideal example of an infinite process.



#187
Nov2206, 07:49 PM

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What does transfinite mean? Boundless but not infinite?



#188
Nov2206, 07:55 PM

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The standard natural numbers form an (external) subset of the hypernatural numbers. We (externally) define that a hypernatural number is transfinite if and only if it is larger than every natural number.
The word "transfinite" is used to distinguish it from the standard usage of "infinite", since, for example, a transfinite sum is something different than an infinite sum. (But their values are infinitessimally close, if the summand is well behaved) 


#189
Nov2206, 10:56 PM

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So basically transfinite numbers are numbers which are larger then any finite number but smaller then infinity?



#190
Nov2206, 11:04 PM

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(resisting urge to go into what is probably unnecessary detail) 


#191
Nov2306, 01:20 AM

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So, you say, that you are going to clean up the mess by adding some more infinite stuff?
Well, maybe, who knows, but currently those transfinite shadow balls are nowhere defined, inside the Newtonian world. It's still to be done, if it's of any use, anyway. Now, it's no solution. 


#192
Nov2306, 03:55 AM

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#193
Nov2306, 04:19 AM

P: 218

Don't steal me my thread, please. Go elsewhere. Unless he somehow solve the paradox I gave, with those hypernaturals. Hypernumbers deserve a new topic. Here are welcome only iff something become clear using them.



#194
Nov2306, 06:20 AM

P: 406

When we write [tex]\sum_{n=0}^{\infty} a_n[/tex], what we mean is; [tex]\lim_{k \to \infty } \sum_{n=0}^{k} a_n[/tex] It's clear as crystal. In each limiting case, there is a leftmost ball and the center of mass remains fixed. In the limit as [tex]k \to \infty[/tex], the accelleration of the center of mass is zero. And that is all we can say without progressing to very esoteric arguments about things like hyperreal numbers etc. 


#195
Nov2306, 06:44 AM

P: 218

You argument is false, OMF.
If it hadn't been, you could just as well proved, that the biggest natural number exists. Bigger than any other. Well, it's a basic mistake on your side, trust me! 


#196
Nov2306, 07:28 AM

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The other approach, the one that leads to the paradox, is, by NOT setting up a sequence of situations with more and more balls, but by considering all balls at once, and calculate the total force on each individual ball. If you do that, it turns out that the sign of the force on each individual ball, by the entire set of all balls, is the same. Adding now the forces of all balls together will then result of course in a net force with the same sign. There are similar situations where you cannot just consider sequences of physical setups, and take the limit of a quantity in this sequence, as the value of the quantity that would occur in the limiting situation. Another example is this: Consider an Euclidean space with a homogeneous, constant mass density. Turns out (by symmetry) that this mass density doesn't result in any (Newtonian) gravitation force on a test mass. However, if you approach this situation by considering a sphere of radius R with homogeneous mass density, and 0 outside, then your test mass will undergo, for each value of R, a specific force towards the center of the sphere. If R > d (distance between test particle and center of sphere), then this force will not change anymore. So the force, as a function of R, grows first, and becomes a constant from the moment R > d. Taking the limit R  > infinity gives you this constant force. Nevertheless, the physical situation with R> infinity is a space filled with a homogeneous mass density, where the force should be 0. 


#197
Nov2306, 08:00 AM

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#198
Nov2306, 08:21 AM

P: 218

> I assure you, an infinite number of non zero numbers sums to infinity.
1/2+1/4+1/8+ ... = 1 Don't you think so? 


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