( A Part 2)
What do we say when we say "This sentence is not strable demo?
In recent years, for various reasons and in different areas, I have had to extensively discuss the proof of Gödel's theorems. On these occasions I have noticed that there are questions that appear repeatedly in the public and, perhaps, be shared by some readers of this blog.
The intention of this series of posts is to try to dispel those doubts. Do not try to develop in detail the proofs of the theorems of Gödel, but doing a general overview with special emphasis on two points:
1. The first call Gödel's Incompleteness Theorem says that, as an axiomatic system for arithmetic to certain conditions (which we recall below), it is always possible to find an arithmetic statement can not be proven nor disproven from those axioms.
often said (I myself have said on more than one occasion) that the proof of this theorem is to construct an arithmetical statement which reads: "This sentence is not provable."
But is it really? Is he really the assertion of its own non-provability? The answer is that no statement speaks for itself. More precisely, we see that the statement may be seen as self-referral accepted only if certain arbitrary conventions that are external to the system of axioms.
The second point to discuss is this:
2. Take, for example, the Peano axioms (which are axioms of arithmetic). Accept that these axioms are a consistent (as in fact, generally accepted). Second-called Gödel's Incompleteness Theorem shows that the statement "The Peano axioms are consistent" can not be proved or disproved from those axioms.
However, if a statement P can not be proved or disproved from an axiom system (call it A) then both P and its negation statement can be added to the system A and in both cases have a system consistent. (The classic historical example is to take A as the first four postulates of Euclid and P, the parallel postulate.)
In particular, this means that the statement "The Peano axioms are consistent no" can be added to the Peano axioms so that the resulting system is consistent!. Now ... Is not it weird? How can it be consistent with the axioms of Peano's statement denying (falsely) that these axioms are consistent? As we shall see, the paradox is only apparent and results from a bad interprertación it really tells the simple statement "The Peano axioms are not consistent."
The task is posed. It remains only roll up and carry it out.
continued ...
( Part A 2 )
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