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.


viernes, 28 de marzo de 2014


By Alfredo Salvador C. García
Mexico City

Victor de Broglie, who proposed
the intrinsic ondulatory and corpuscular
nature of all particles.

Let be the following question:

«Can we explain photons as waves traveling that, moving at speed of light, their wave lengths would be contracted (Lorentz contraction) until infinitely small packets of ondulatory quantized wave are obtained, relatively to observers slower than it?»

Using Logical Mechanics reasonings [this name has been proposed for Quantum Mechanics (in general, for Physics) being studied by its axioms, laws and theorems researching their logical nature], there is:

1. pphoton=h/λphoton that is De Broglie's hypothesis (with λphoton the wave length of travelling light, and h Planck's constant).
2. pphoton=h/λphoton=h·fphoton/c, because c=λphoton·fphoton (c speed of light, and fphoton the travelling light wave frecuency).
3. If according to Planck's energy quantization Ephoton=h·fphoton, then Ephoton/c=h/λphoton.
4. λphoton·Ephoton=h·c is valid for a rest state.
5. λphoton'·Ephoton'=h·c is valid for movement state.
6. λphoton·Ephotonphoton'·Ephoton', and if movement happens at light's speed c,
7. λphoton·Ephoton=0 is deduced, because λphoton=0 agreeing with contraction described by Special Theory of Relativity (Lorentz' contraction that was mentioned), and
8. λphoton'·Ephoton'=0 is deduced necessarily (from propositions 6 and 7).

Can Ephoton'=0 be? May be, but it would be necessary to develop any experimental measurements that, in principle, can not occur because 1) it is impossible that a rest state experiment would become a speed of light frame experiment; and 2) because it is impossible to be indeed a photon (they are the only entities that could “tell us” everything about observations made at speed of light c). The same analysis can be done trying to explain if λphoton'=0 do is recognised true.

9. λphoton'·Ephoton'=0 is deduced undecidable (even being true, it can not be deduced if it is either true or false: if it were true, an experiment made at light's speed would occur, but that is false; if it were not true, the speed of the frame where the experiment is built would not be speed of light c, then it could not be measured if really λphoton'·Ephoton'=0 happens or not –because it is supposed that proposition is evaluated travelling at speed of light–).
10. λphoton·Ephoton=0 can not be deduced, because λphoton'·Ephoton'=0 being undecidable can not be a valid proposition for the present deduction that pretends to be coherent (because it is undecidable and can not express any kind of truth). On the other hand, the proposition λphoton·Ephotonphoton'·Ephoton'=h·c can be deduced and is valid only if the frame of reference is not intended to be moving at speed of light. If the contrary were, undecidable propositions would appear in this deduction as true and, consequently, would imply following absurd propositions.

Therefore, it can not be explained, in principle, if «infinitely small packets of ondulatory quantized wave» would be either observed or not.

In conclusion, the initial question can not (and must not –if it is supposed that Physics and Logic theorems and laws are followed–) be answered. Even if any satisfactory answer existed, it would not be found either as true or as false (it would not be known if it do is a correct or an incorrect answer).

March 28th , 2014

Note: The question at the top was originally proposed, as it is in the draft, by Ferrer, K. This phenomenon was infered by Einstein since he was 16, and nowadays Ferrer's action has inspired the present deduction. Alfredo Salvador C. García.

1 comentario:

  1. Additional note: The question as it is structured (with some little modifications) was originally proposed by Ferrer: https://www.facebook.com/EverybodysGotSomethingtoHideExceptMeandMyMonkey?fref=ufi This idea was proposed also by Einstein since he was 16.