Teorema de incompletitud de Gödel

Kurt F. Gödel, en «Sobre las proposiciones formalmente indecidibles de los Principia Mathematica y sistemas afines» [paráfrasis]:

«Existen argumentos lógicos imposibles de ser deducidos verdaderos o falsos; entre ellos, la coherencia de dichos razonamientos.»

La existencia verdadera o falsa de algo (por ejemplo, las piedras; al contrario, las hadas), no implica que la misma sea demostrable así, ni que deba o no tenerse fe en cualquiera de estas posibilidades.


La creatividad surge de hallar –pensando diferente del resto– ideas absurdas, para así nuevamente pensarlas y darles coherencia.

Ahí la importancia de la Lógica: porque sólo con ella es posible tanto hallar los absurdos como obtener la coherencia.


sábado, 22 de marzo de 2014


By Alfredo Salvador C. García
Mexico City

Erwin Schrödinger, who took
the ondulatory nature in order to describe
corpuscles movement.

Let be Ψ the function that describes correctly the movement circumstances (or movement state) of particles. If this is true, it will be coherent with all laws describing the Universe.

So then, Ψ(x,y,z,t) is expressed because movement circumstances of particles are given by the position (x,y,z) and the time coordinate associated to them (t). In that case Ψ describes both position and time for particles movement.

Supposing that Ψ=f(x,y,z), where f(x,y,z) is any position function, Ψ is independent of the position's localization; its movement circumstances are valid to any position in the Universe, or what is the same, because, in principle, the movement circumstances of a particle are intrinsic to it, the particle may be found at every position in the Universe necessarily; the circumstances associated to the particles movement are spacially present for all over the Universe. In other words, the particle is a wave that is spacially and constantly distributed through the Universe.

To research if this is true, it is just necessary to localize at any region of the Universe another particle with its own Ψm value (m for measurement; the particle is used to measure). When a change is presented for the Ψm value, it will be known that the first particle was found in a determinated place. After be done in many times at different places, it will be observed the same variation for the Ψm value. However, doing this also modifies the Ψ value and does not allow to determine if it actually makes reference to the particle whose circumstances were proposed for the original Ψ value. This is, if Ψ changes its value, it does not describe any more the movement circumstances of the particle in the same way it is desired to be known, with the original value of Ψ.

To reach the Ψ value for a particle in some movement circumstances so any particle can not have influence to another particle in its value is not possible. For that it is required to do a measurement, and that means to modify the Ψ value and, eventually, not to find the Ψ value in the required circumstances. This occurs, but may be the Ψ would be found because it has influence by itself and determines its own Ψ value that would not change: the movement circumstances does not change because there is not another particle that modify them. How could be a particle having influence by itself? Note the following reasoning: Ψ only changes its value because of a particle having influence to another particle. Then, if Ψ does not change its value, it can be explained by the lack of influence. Only a particle can have influence to another particle and change its Ψ value and its movement circumstances. That is why a particle can not have influence by itself; Ψ can not change its own value, in principle, in order to its corresponding particle would find it.

Consequently, Ψ=f(x,y,z) is actually false if it is assumed for a particle without influence from other particles, because Ψ can not be known according the proposed circumstances. In general, any Ψ value is false by the same reasoning.

Schrödinger's equation proposes the same, let us say, a Ψ=f(x,y,z) value known for the indicated circumstances, then this equation is false for this condition. What is, it can not be valid to use Schrödinger's equation for particles that are not having influence, evidently, by other particles. It does not mean that the equation is not valid for other circumstances where, in fact, it has been proved completely valid. Only for particles that do not have influence has not been proved if it is or is not valid. Note that it has been proved the case where a particle is not having influence and then it reaches some influence (by the Ψm initially proposed), but never for a particle simply without influence. Cases where there is a change from the lack of influence to the influence are various and known: any kind of «tunnel effect» situation, the Heisenberg's collision between a photon and an electron –that derives as a result its uncertainty principle–, and some others. It has never been observed the case of a particle perceiving itself.

This implies that Schrödinger's equation is only valid for circumstances where there is not interaction between particles, also called «measurement». Then:


that is Schrödinger's equation with h Planck's constant for energy quantisize, U the potential energy, and E the total energy associated to the particle, may be modified for the circumstances that can not describe. It is proposed a new equation as follows:


where 2·J is the «importance factor» defined as

J={0 if there is not a measurement; 1/2 if there is a measurement.

Is called 2·J «importance factor» because it indicates if it is necessary to quantisize or not the energy. If J=1/2, h value becomes necessary (or important) to the equation, also as the quantization that this constant implies. If J=0, the h value is not needed in the equation (it does not change its value, but J=0 makes us ignore it) and the energy quantization becomes not important. This 2·J importance factor is also present in the redefined Heisenberg's uncertainty relation, let us say,

Δr·Δpr=J·h/(2·π), with r=x, or r=y, or r=z.

When a particle has influence due to another particle, J=1/2 and the Heisenberg's uncertainty relation is obtained in its original form. When there is not influence in a particle due to another particle, J=0 and the uncertainty does not have any physical sense; the particle is necessarily determined by its position and its moment (or velocity). Because Δr·Δpr=0 implies Δr=0, that the particle is determined by its position at r: the particle is at the same place where it is (does not matter it sounds tautological) even if its nature is ondulatory or corpuscular, or another kind of nature, and this agrees the fact that Ψ does not change its value. Furthermore, Δpr=0 is also valid because the particle is moving at the same velocity itself is moving (does not matter it also sounds tautological), and this agrees again the fact that Ψ does not change its value: if the reference frame is found from the same particle's perspective, p=0 in any case and Ψ effectively does not change its value.

Now it will be proposed again the solution to Schrödinger's equation for the particle without influence. If this happens, J=0. Also, U(x,y,z)=0 because there is a lack of influence from another particle; there are not forces generating any kind of potential energy. Finally, if the particle is not moving according to a frame following the particle's velocity and there is not potential energy, it is assumed that the total energy associated to the particle is E=0. Therefore:


is found without considering any Ψ value, just as it was specified, let us say, that any known Ψ value was false. By this it can be concluded:

1. If n=1, where n is the number of particles in a reference frame, J=0.
2. If n is greater than 1, J=1/2 (this implies an interaction between particles). In general,

J={0 if n=1; 1/2 if n is greater than 1.

3. Again, 2·J=0 indicates that energy quantization is not important; 2·J=1 indicates that energy quantization is necessary.
4. Schrödinger's equation is redefined as


and the uncertainty relation becomes

Δr·Δpr=J·h/(2·π), with r=x, or r=y, or r=z.

5. Even if Ψ is not determined for its value when n=1, Δr=0 and Δpr=0 are according to the only phenomenological agent, the studied particle. Has Ψ a value whose meaning is for the rest of the particles in the Universe; for the studied particle, Ψ is also not important in its value as the energy quantization is not. That is, according to the particle, the only fact of being a particle is sufficient to express its movement circumstances; the particle does not require a Ψ value, or another property value, to express its movement circumstances by itself.

Consider that the influence of a particle by another, or an interaction, or a measurement, implies the field particles exchange. Therefore, n=1 when a particle is not exchanging field particles, or what is the same, when it does not interact with any particle. Then, n=1 for the particle until any region where there are not any particles (this sound repeatitive, but is necessary to be mentioned to observe that coherence exists between the phenomenon description and the used simbols). If n=0, J is not defined; this implies that either J=0 or J=1/2 are the same valid and the same obtain contradictory conclusions for the space-time nature as a physical quantity (n=0 expresses that only the space-time is analysed). This entity can be studied by other reasonings (it is necessary to have other conditions different to the described by Quantum Mechanics), by the use of Relativity Theory.

March 22nd, 2014

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