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.

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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.

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lunes, 13 de enero de 2014

ON THE MEASUREMENTS AND THEIR DETERMINISTIC CONSEQUENCES


By Alfredo Salvador C. García
Mexico City

Gödel

ABSTRACT

Where the deterministic consequences of measurments solve the schism in Physics, between Quantum Mechanics and Relativity Theory.



I. INTUITIVE MEASUREMENT DEFINITION

Let be an object. Then, there is a hypothetical “box” where it is introduced. The box measures a characteristic of the object, a property. If the object fulfills the property, the box will show as a result in a screen or a panel, whatever it is, a «yes». On the contrary, if the object does not fulfill the property, the box will show a «no». This intuitive reasoning outline is called measurement. Therefore, to measure is to recognize a property from an object.

The reasoning can be symbolized in order to formalize it. Then, if f(P)=yes, the object P fulfills f(P), the property of that object. On the contrary, if f(P)=no, it means that the object does not fulfill the property. An example of measurement is the following:

There is a 15 cm stick. It is introduced in a imaginary box of 15 cm height exactly as the ruler that is joined to it marks.

Actually, the ruler has a great variety of imaginary boxes that crosses it, but that is not relevant for the measurement: the stick is introduced and fits tight the box, so the latter tells automatically to the brain «yes». That the stick do measures 15 cm.

Another measurement example is the following:

There is a soccer ball. This is introduced in an imaginary box that tells to the brain if the object inside is either spherical, or not. For the case, the box tells to the brain «yes».

It is not questioned right now which is the precision that the boxes have in both examples. In other words, if it is assumed that the boxes can make mistakes of their veredicts, then the precision is the probability of good aim. This probability is determined by the user of the measuring box. How a “good design” is confered to the boxes that the user would built with his criteria designates directly the precision of the measurements. Even when the measuring boxes are real, if the user (or the manufacturer that sells the box to the user) confers it either these or those criteria at the moment that the measurement occurs, then they will influence directly the precision of its measurements. That is, some people could say that the ball is not spherical because their boxes have a higher precision than the rest, whose boxes have a precision that is, evidently, lower.


II. UNCERTAINTY PRINCIPLE

Let us suppose there is a box that measures the precision of other measuring boxes. The corresponding symbol of this box is π( ). If the measured box presents a precision of 1, that is, never makes mistakes of its veredicts, then the box that measures will show a «yes». If the contrary situation occurs, that the box presents a precision lower than 1, then it will show a «no».

Again, when the K box is introduced to the measuring precision of 1 box (MPB1.0), it can be possible either π(K)=yes, or π(K)=no. What happens if a MPB1.0 identical in all of its aspects to the MPB1.0 evaluating the K box is introduced to the latter? Two options are possible:

1. π(MPB1.0)=yes. That means that both MPB's would be of precision 1, because they are identical in all of their aspects.
2. π(MPB1.0)=no.That means that both MPB's would be of precision lower than 1, because they are identical in all of their aspects.

If the second case occurs, it exists the possibility that the MPB1.0 measuring would have made a mistake and, as a consequence, we would not have the full certainty that the MPB's, neither of both –what is said about one is necessarily said about the other because they are identical in all of their aspects–, are of precision lower than 1. Nevertheless, neither the possibility of a precision 1 could be, because there is a «no» as an answer that would show that, actually, neither of both MPB's have a precision of 1. It is not relevant the number of measurements done: it is impossible that the MPB's in the second option would have any kind of precision.

If the first case occurs, π(MPB1.0)=yes, then there would not be any certainty that the MPB1.0 measuring has either a precision of 1 or not, because it exists the possibility that its answer were a mistake. Naturally, this talks about both MPB's. Neither the number of measurements is relevant: there is always the possibility that the «yes» were a mistake. If it were, the second option occurs. Therefore, even with the MPB's answers known, it is true the following

Theorem. «It is impossible that any MPB1.0 had any kind of precision.»

Then, either the π(K)=yes formula, or the π(K)=no formula, with K any box that is not a MPB1.0, would be no only questionable, but absurd. That is, if a user can not give any precision to the box, like the MPB's, is because the box is not measuring anything. It is impossible to the box. The argument for this is based on the measurement concept that implicates it necessarily, it has been shown with examples, the precision concept. If the box can not have any kind of precision is because no measurement can be done that permits the evaluation about the number of good aims and the number of mistakes. If the measurements were possible, it is not relevant the precision quantity, these can be made and permit the declaration of a quantity. Being impossible to declare a precision value, the box refered does not measure.

It could be suggested, on the contrary of the MPB's, that a pair of mutually identical boxes could measure if other boxes have either a precision of 0, or not. Nevertheless, this would be equivalent to have MPB's measuring a precision of 1: if the answer of the suggested boxes is «no», it would be equivalent to have a «yes» from the MPB's measuring a precision of 1. The same if the answer is «yes», it would be equivalent to a «no» of the MPB's measuring a precision of 1. To convert the suggested boxes (being them MPB0.0) into MPB1.0 ones is only a question of an attachment that even could be mental. Then, it is announced the following

Uncertainty principle. «The precision can not be neither 0, nor 1 in any measurement.»

If the Theorem is followed, the MPB's does not exist due to their lack of precision and it can not be guaranteed that the precision of other boxes that are not MPB's value either 0, or 1. The only way to get a precision value would be by measuring in many times and declaring if either a good aim is obtained, or a mistake, in each one of them. Being possible to declare the good aims until a k measurement, nothing makes sure, as a principle, that the k+1 measurement is not a mistake. The uncertainty (possibility of making a mistake) is an intrinsic question of measurements.


III. DETERMINATION PRINCIPLE

Let be a P object with a j property that shows the not-j property (a property that is not j) when is introduced in any measurement box.

Then, the P object is introduced into a box that measures if the j property is shown or not, assuming that the measurement describes the object. As it is expected, the box answers «no». What is more, the box can answer «no» in as much times as it is desired.

Firstly, the uncertainty principle implicates that even getting in many times the «no» as an answer, it exists the possibility that it were a mistake. Then it can be suggested that

1. the P object has the j property even when the measurement happens.
2. that the object has not the j property, but it has the not-j property.

Both are valid and coherent with respect to the uncertainty principle. The first implicates that the «no» of the box is a mistake. This is permited by that principle (the precision can not be 1). The second implicates that the «no» is a possible good aim. This is also permited by the same principles (the precision can not be 0).

If the first proposition is observed, it means that P has the j property because it is P, but it shows the not-j property because it is measured.

If the second proposition is observed, it means that P has the j property because it is not measured, and when it is measured it has the not-j property because it is shown.

Nevertheless, P, to continue being P, should not modificate its properties. That is why the verb «to show» is used in the definition of P. That the P object shows not-j does not implicate necessarily that it has not-j. Therefore, it is assumed the following

Determination principle. «Any object is not modified in its properties, not mattering the circumstances.»

From the same P, the fact of showing not-j when it is measured is one of its properties. If j were «to show not-j», it would implicate that P will show j (because not-j is shown and is one of the P properties) and, at the same time, that will show not-j (because j is shown, that is, «to show not-j»), and it is absurd. By this way the determination principle makes sure the possibility of the existence of objects like P that do not modify its properties, but that do modifies the presentation of them for any measurement. The determination principle makes sure that j is not a property like «to show not-j»: if P can not modify its properties, and «to show not-j» is one of them, j can not be «to show not-j» because this modifies the property «to show not-j». That j is «to show not-j» does not respect the determination principle. The j property can be only «to show not-j when P is measured».


IV. PHYSICS MEASUREMENTS. RELATIVITY OF THE WAVE-CORPUSCLE COMPLEMENTARITY

The principles that make reference to the measurements are coherent between them and that is why they will be taken to analyse the Physics measurements; the intention is to solve some of the Quantum Mechanics questions that are aparently contradictory with respect to the Relativity Theory of A. Einstein, or, what is more intriguing since the formulation of the Quantum Mechanics, that seem not to have a common sense interpretation.

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The Heisenberg's uncertainty principle indicates that it is impossible to measure with a precision of 1 and simultaneously both the velocity and the position of an object in the time-space. This, surely, is compatible (does not reach any contradiction) with the uncertainty principle expounded here. Nevertheless, the interpretations made about it are not compatible with the determination principle.

Concretely, when a photon pushes another particle, both the velocity and the position of the particle are measured. The interpretation nowadays accepted is that the particle gets itself “diffused” probabilistically through the time-space when the collision (the interaction between the photon and the particle) occurs. Then, it is impossible to determine with a precision of 1 both the particle's position and the particle's velocity simultaneously. However, Einstein observed that the particle would not be fulfilling the determination principle: it leaves its corpuscular nature and gets an ondulatory one (a probabilistic wave for the position and the velocity of the particle is formed), so that implicates a paradox endeed. The particle is exchanging its corpuscular properties for some ondulatory.

The solution to this aparent paradox is based on the following reasoning: in the same way the j property was for P even if the not-j was shown, a particle continues being corpuscular in its position and its velocity to itself, but it is shown as a probabilistic wave in the time-space to the detectors (that measures the particle).

The phrase «the particle is to itself» means that in the case we as observers took the place of the particle (if we were the particle), we would not find any change in its nature. The presentation change to the detectors not only would seem normal, but inescapably necessary. On the contrary, the perspective being us only observers makes them measure that the position and the velocity are shown as probabilistic waves. This comes to support the interpretation given by N. Bohr, affirming that both ondulatory and corpuscular natures of a particle were only «both sides of the same coin», and that both characteristics were complementary endeed. The missing aspect to be cleared was the relativity of both perspectives: who measures the probable characteristic of a particle (like the position), percibes it as a wave; who can not measure that property (like the particle –it can not measure itself, in principle–), it is shown as a corpuscle. Of course, was the invarancy of the charge and the angular moment of the particles that thing allowing the assumption of the diffusion of them percibed by the detectors. Those properties of the particle are not modified: only the velocity and the position change, these properties relative to the reference frame where they are measured.


V. PHYSICS MEASUREMENTS. EPR PARADOX SOLUTION.

It is not strange by this way that the EPR paradox would be solved as a consequence. Suggested by Einstein, Podolsky and Rosen, talks about the interaction (collision) of two electrons that are propagated as probabilistic waves, not only for their position or their time coordinates too, but for their spins, when they stop interacting. While they were interacting, their spins were different. The spins, according to the Quantum Mechanics, of a pair of electrons that are interacting can not be the same (W. Pauli's exclusion principle). The electrons only have two options for their spins: either it is 1/2, or it is -1/2. During the collision, if an electron has a 1/2 spin, the other has necessarily a -1/2 spin. However, when an electron is independent of another electron, both in their ondulatory form, neither of both spins are known because their are not measured.

The scientists reasoned that being those electrons mutually independent after the collision would implicate the lack of motives for their mutually dependece. The Quantum Mechanics deduces that, on the contrary, the spins of the electrons are complementary all the time and if the measurement of one of them is known, the spin of the other is the other of both spin values possible. The measurements confirm this: the electrons are, in all the cases at the moment observed, related by their spins. This is, even they had already interacted, the electrons continue mutually related by their spins as if they were still interacting.

Again, Einstein (that never agreed with the intuitive meaning interpretations about the Quantum Mechanics, according to the common sense) shows that the determination principle is not respected. Being the electrons independent, the collision has already happened, they should have each one independent spins, a characteristic of the also independency of the electrons. If the measurements done have as a result that the spins are dependent is due to the following: when the spins of the electrons are not measured that property becomes ondulatory with respect to the observers that could measure the electrons in their corpuscular characteristics, those invariant; with respect to the electrons, their spins are the same since the collision finished. For the electrons (that does not make, and can not make, any measurement) the spins do not get an ondulatory form, probabilistic, and they are conserved just like they are, but their presentation to the detectors do get it. For the measurers of the pair, it is only observed a spin probabilistic wave and being detected as corpuscles, the spins seem related, as if the electrons continued interacting.

Therefore, after the collision the electrons do not interact indeed, but that perspective can be percibed only by the electrons themselves; the observers can not distinguish with the ondulatory spins if the electrons continue interacting or not and could assume that they do it. To assume also that with respect to the electrons the spins get a probabilistic form is equivalent to assume that the previously defined P object showed the property not-j because it had it. It was observed that consideration was compatible with the uncertainty principle; this was the cause of the difficulty at the moment of trying to solve the EPR paradox, that is, that the electrons do were ondulatory (talking about the spins, positions and velocities) with respect to them.

One might say that the electron not interacting is not capable to percibe how the other electron spin is: they also mutually percibe them as probabilistic particles. In that way, it is impossible that they could interact, because they are not do interacting: the electrons are mutually probabilistic waves due to this lack of interactivity. The electron could find the same strange as Einstein found this fact, but it is justified by the existence of the determination principle as it has been exposed. The electron percibing its counter-electron is presented to itself as a corpuscle, and percibes its counter-electron as an uncertain object. The counter-electron is presented to itself as a corpuscle and percibes the first electron as an uncertain object. Both electrons are corpuscles of unchangeable complementary spins (due to the determination principle) even they are independent since collision ended, no observation can measure it with a full certainty (due to the uncertainty principle).


VI. PHYSICS MEASUREMENTS. UNION OF QUANTUM MECHANICS AND RELATIVITY THEORY.

The last pair of sections about physics measurements confirm that either the intepretations similar to Bohr, or similar to Einstein were only, the same as the particles being waves and corspuscles, the obverse and reverse of the same Physics. Particularly, the EPR paradox showed a clear contradiction not only with respect to the determination principle, but also with respect to the light's velocity invariance principle that implicates, as it is known at this moment, that it is the top permited in the Universe for moving objects. If two independent electrons could mutually determine their spins simultaneously (as if they were suddenly interacting), the information of one spin value would be transmited immediately, with an “inifinite”velocity –higher than light's velocity–. That is the reason Einstein had to not find coherence between his own Relativity Theory and the Quantum Mechanics, both correct according to the measurements made to confirm them.

Then, both theories are, in fact, complementary parts of the same Physics. By one side, Quantum Mechanics describe how are the objects measured (and how do they behave) with respect to who are measuring –or can measure–. The Relativity Theory explains how are the objects observed (and also how do they behave) with respect to themselves. The formulation of the uncertainty and determination principles let us find the link between these two theories, the most representative of nowadays Physics and, at the same time, let unite them. For example, with that proposal it is easy to explain why Relativity Theory seems to have an absolute dominance over the big (massive) objects description, while Quantum Mechanics seems to be prohibited for that description.

That is because a particle has the ambivalent characteristic that determines it, to be wave when it is observed and to be a corpuscle with respect to itself. If the number of particles increases, the uncertainty over the position and the velocity measured together decreases. If this happens, all of them are manifested less ondulatory with respect to the detectors and, therefore, more corpuscular. In that way it is, that a common size object, compounded by a various number of particles, is not only observed corpuscular to itself, but also with respect to the observers. Then, the corpuscular nature seems to dominate for massive objects and because this nature is described by the Relativity Theory exclusively, it seems to be valid only for the explanation of massive objects, but actually it is valid in all the corpuscular cases, either massive objects, or particles manifested to themselves. Furthermore, Quantum Mechanics is valid in all the cases where the objects are presented as waves, that is, objects whose position and velocity uncertainty (and any property indeed) is “high”, for example, not polymeric molecules, atoms, chemical complexes, etc., and specifically for particles presented to the detectors.

The junctions previously made relating both theories move, without showing it evidently, between the perspective of particles and the perspective of detectors. One of these cases is the deduction of the anti-particles existence made by P. Dirac. It assumes, in broad strokes, that the energy is quantisized, and that a particle that passes from a negative energy region to a possitive energy region leaves a “pit” in the side with negative energy that corresponds to the anti-particle of the particle moved. A particle can only pass from one region to another because it turns into a wave: so, the particle can be found with an uncertainty value in the positive energy region and another value of uncertainty in the negative energy region; later, the particle gets a higher certainty in the region whose energy is the opposite of the original position energy and, eventually, in that region the particle gets its corpuscular nature again. Its velocity during the phenomenon can not overflow light's velocity. And that is what really happens: the particles and the anti-particles exist and have been observed in various particles reactions.

Dirac's perspective moved from the corpuscular form to respect the light's velocity limit to the ondulatory form in order to allow the movement between regions with different types of energy. What the particle would observe during its movement from a region to another is: it measures the energy in the regions, so this can not be uniform all over the space, but a wave of uncertainty through it. Then, the particle can move without difficulty between both regions because the uncertainty let it “doubt” if the energy is really there, until it gets its last position in the region with opposite energy. With respect to the particle, it never changed into an ondulatory nature; the energy in both regions did it. This do not contradict in any instant the Relativity Theory, because the particle is moving with a velocity that does not overflow light's velocity. Dirac's arguments allow the conderation of the particles as waves and corpuscles simultaneously, then it is possible to relate both Quantum Mechanics and Relativity Theory getting results that are consistent with experimental reality.


VII. CONCLUSIONS

From the intrinsic measurement principles it could be possible to solve the intuitive questions that caused a great difficulty since the creation of Quantum Mechanics. The aparent paradoxes observed arised from the paradigm contradicting the determination principle. Not to have it formally defined was the reason of the assumption that every object showed some characteristics for the measurements because they had them, but this precisely reaches contradictions not only for the Physics, but also to the elementary Logic (that P shows as properties both j and not-j).

We can still trust that both Physics theories are valid to describe reality. It is only a question of interpreting them as corresponds: Quantum Mechanics for ondulatory presentations and Relativity Theory for corpuscular presentations. Both are related by the objects that observe, but the form they observe them make appear the differences.

January 12th, 2014


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