The Gödel's theorems aim to found logic on an axiomatic basis which is out of reach.
Whatever system of axioms is used to build a theory, there are propositions that we know to be true but whose the truth cannot be demonstrated within the framework of the system.
The axiom in a theory is a basic formula that is considered true without proof.
The inconsistency is to be able to demonstrate one thing and its opposite.
incompleteness characterizes truths mathematics that cannot be proven.
Whatever the richness of a system of axioms this cannot match the capacity of the potential content of thought.
explicit thinking – result of our reflections based on a finite number of axioms – is simpler thancomplex thinking which in theory cannot realize.
To get out of the dilemma of a true and wrong at the same time, you have to get out of the system itself, get into meta position, an external vision, by adopting a broader system.
Logic has its limits ; in any system there are indemonstrable truths.
Any finite set of sufficiently rich axioms necessarily leads to results that are either undecidable, either contradictory.
Any human logical system is incomplete if it wants consistent. Coherence requires incompleteness.
The condition of incompleteness encountered by the scientist is not a defeat of reason but a chance to progress in introducing him to the confrontation with mystery, to the mystery of knowing.
Einstein's Formula, ” most incomprehensible, is that the world is understandable “, and setting evidence of the ” fertility ” of incompleteness are like two ” signs ” of the mystery of knowing in the modern scientific approach.
The truth cannot be expressed in terms of demonstrability.A provable thing is not necessarily true and a true thing not necessarily provable.
To find truths in a given system it must be able to get out of it and for that have a reason capable of creating a system in which the old indemonstrable truth will become quite demonstrable.
The scope of Gödel's theorems matters considerable for any modern theory of knowledge. First of all he does not only concern the field of arithmetic, but also all mathematics that includes arithmetic. But mathematics, which is the tool of basis of theoretical physics contains, obviously, Arithmetic. That means that any comprehensive search for a physical theory is illusory. If this statement is true for the domains most rigorous study of natural systems, how could we not dream of a complete theory in an infinitely more complex domain – that of social science ?
The Gödelian structure of the set of levels of reality, associated with logic ofthird parties included, implies the possibility of constructing a complete theory for describe the transition from one level to another and, a fortiori, to describe all levels of reality .
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