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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martes, 11 de diciembre de 2012

INDUCTIVE DEMONSTRATION FOR GÖDEL'S THEOREM

It is considered that consistent formal systems can be constituted by a sequence of propositions that are associated, at least the axioms, each one with an object corresponding to a suggested model. So, any proposition can have its own object that also belongs to the universe of the model if it is deduced from the axioms. Eventually, some rules of inference that allow the creation of new propositions that relate the objects of the universe of the model implicate directly the demonstration of Gödel's theorem.

Specifically, there are proposition kind A that are structured as it follows:

For all F, R(F,M(F)).

And R(a,b) is a relation defined to admit the object M(F) (that belongs to the universe of the model).

This proposition A1 can have its own proposition A2 that relates its object M(A1) with the former. Doing this by induction, we obtain a proposition An. Then, for the proposition An+1 the process of obtaining its own proposition that relates M(An+1) with An+1 is stopped. This deduces directly that it does not exist a proposition An+2, because the process was supposed stopped for An+1. If it is not related with an object M(An+1), it can be proved neither true, nor false. It does not have any opportunity of assesment (either truth or falsehood) that could be included for the unexistant M(An+1). From this, the conclusion is automatically recognized:

I. For every consistent formal system there exist propositions that are undecidable (neither true, nor false).

Actually, it can be observed a second conclusion:

II. Every consistent formal system can not be proved consistent by itself. Every consistent formal system is necessarily incomplete, because all their propositions (axioms and theorems) can be proved neither true, nor false.

Note that «The consistency of the formal system» is an undecidable theorem of it.

July 2012


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