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April 10th, 2017, 01:52 AM  #1 
Newbie Joined: Jul 2012 Posts: 5 Thanks: 0  Inconsistency in set theory. A while ago I’ve send following article where I describe this inconsistency to about 100 mathematicians from different universities to get some feedback. http://www.ardix.be/articles/The_abs...ero_vector.pdf The strange thing is that the reaction is nearly always complete silence. A few where angry or not to the point. So I assume the article is correct, but I still have completely no idea what the silent majority of mathematicians really think about it. Hopefully people on this site have more to tell about this. 
April 10th, 2017, 02:18 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,967 Thanks: 1607  Such a formula would be for the resultant of a set of forces, which is not the same thing as the set of forces.
Last edited by skipjack; April 10th, 2017 at 08:13 PM. 
April 10th, 2017, 08:52 AM  #3 
Senior Member Joined: Aug 2012 Posts: 1,891 Thanks: 526 
I looked at your link. I see the problem. You start by saying that if no forces are acting on the object, the force can be represented by $\emptyset$. Then you say that if no forces are acting, it's represented by the zero vector. Then you say AHA! Since the empty set isn't the same as the zero vector, set theory is inconsistent. Isn't the real problem simply that you chose to notate the same thing using two notatiions that aren't the same? 
April 10th, 2017, 08:54 AM  #4  
Newbie Joined: Jul 2012 Posts: 5 Thanks: 0  Quote:
Even in case there is one force the resultant (in the same point of application) is the same as the force. In the future, I will also write “force acting on the center of mass” that looks a bit more complex, but then at least no one can start as discussion about torque etc., which is in fact completely irrelevant for this topic. Last edited by skipjack; April 10th, 2017 at 08:15 PM.  
April 10th, 2017, 10:32 AM  #5 
Senior Member Joined: Aug 2012 Posts: 1,891 Thanks: 526 
The identity element in any vector space is $0$, not the empty set. End of story. You simply claimed it's the empty set, then claimed it's zero, and concluded that there's a contradiction. The only contradiction is in your own claims.

April 10th, 2017, 10:46 AM  #6 
Senior Member Joined: Jun 2015 From: England Posts: 824 Thanks: 244 
Haven't you forgotten Varignon's theorem?

April 10th, 2017, 11:00 AM  #7 
Senior Member Joined: Aug 2012 Posts: 1,891 Thanks: 526  Me or the OP? I never heard of it, but when I looked it up it reminded me of the geometric proof that the sum of the $n$th complex roots of $1$ for $n> 1$ must be zero. That's because they're points equally spaced about the unit circle. They can be interpreted as vectors symmetric about the origin so their sum cancels out to zero. The corresponding algebraic proof is that $z^n = 1 \implies (z1)(z^{n1} + \dots + 1) = 0 \implies z = 1$ or $z^{n1} + \dots + 1 = 0$. Last edited by Maschke; April 10th, 2017 at 11:11 AM. 
April 10th, 2017, 11:06 AM  #8  
Senior Member Joined: Jun 2015 From: England Posts: 824 Thanks: 244  Quote:
Varignon was the first to realise that a bunch of forces can have zero resultant but still exert a net moment. Incidentally a more fundamental question is How do three component forces which interact form the elements of a set? The elements are supposed to be distinct. If they interact and combine to form one new resultant does the original 'set' still exist? Last edited by studiot; April 10th, 2017 at 11:10 AM.  
April 10th, 2017, 11:13 AM  #9  
Senior Member Joined: Aug 2012 Posts: 1,891 Thanks: 526  Quote:
Thanks for the history tidbit. Hadn't heard of Varignon but it seems he was ahead of Newton on the subject of vectors. ps  According to Wiki, Varignon published his theorem in 1687. That's the same year as the publication of Newton's Principia so they probably came to this idea independently. https://en.wikipedia.org/wiki/Varign...rem_(mechanics) Last edited by Maschke; April 10th, 2017 at 11:18 AM.  
April 10th, 2017, 11:54 AM  #10 
Senior Member Joined: Jun 2015 From: England Posts: 824 Thanks: 244 
The correct use of terminology for turning effects is a pet subject with me. Sorry if it bores you, here is something I posted recently for someone else. [aside] So it would seem to me highly sensible and desirable to identify each of these effects with a unique name. Unfortunately too many mix up the available different terms to create the general confusion, particularly for beginners, that holds today. So Perhaps a little history might help? Ca 250 BC The mechanics of turning effects was known to the ancient world for example the principle of levers attributed to Archimedes. 1725 The term moment was introduced and formally defined by Varignon in his book 'Nouvelle Mechanique.' “The moment of a force, P, about a point O is defined as the product of that force into the perpendicular OM drawn to its line of action from O, this perpendicular being reckoned positive or negative according as it lies to the left or right of the direction of P." Varignon's theorem holds to this day and may be found on Wikipedia. 1750 – 1804 St Vennant investigated the torsion of prismatic bars and posed St Vennant’s Problem. He did not however introduce new concepts in turning. 1804 1806 Poinsot published his book 'Elements de Statique' and the theorem that bears his name. This introduced two things. He defined and introduced the term ‘couple’ and the theorem which states that in 3 dimensions any system of forces may be reduced to a single force plus a couple, in a plane perpendicular to the line of action of the force. He clearly defined his couple to exist in a plane. 1912 Lamb, one of the most prominent applied mathematicians of his time, proposed that the term ‘torque’ be introduced to replace ‘couple’ Lamb 'Statics' p52. “Since a couple in a given plane is for the purposes of pure statics sufficiently defined by its moment, it has been proposed to introduce a name torque or twisting effect which shall be free from the irrelevant suggestion of two particular forces.” This suggestion was not, however generally adopted. Indeed, the three most influential texts (in this subject) of that era and since carried on as before. 1926 Love ‘A Treatise on the Mathematical Theory of Elasticity’ 1936 Southwell ‘Theory of Elasticity’ These both refer to ‘Torsional Couples’ for the 3D effects described in St Vennant’s Problem. 1934 Timoshenko published the third standard text, ‘Theory of Elasticity’ and clearly established torque in this 3D role. In fact, most authors in the second part of the 20th century have followed the notation set by Timoshenko in elasticity. I haven't ventured beyond the first half of the 20th century because nothing new has been added since. It does bring out one other source of confusion. The difference between twist and turn, which is even less often correctly stated. I usually try to associate the Ts Torque, Torsion and Twist. Edited by studiot, 30 March 2017  02:15 PM. [/aside] Last edited by skipjack; April 10th, 2017 at 08:27 PM. 

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