( A Part 7 - Part 9 A )
Gödel's paradox
El Segundo Gödel's Incompleteness Theorem says that if we have a system of axioms as they have Kurt and David, then the claim that this system is consistent can not be proven nor refuted from the axioms of the system. That is, this statement is undecidable for the system.
One consequence of this is that the system of axioms is consistent with the statement that says that the system of axioms is inconsistent! . If the system we add the statement that says "The system is inconsistent" also continue to have a system that is consistent .. How can this paradox be resolved?
To begin, note that our language only allows write arithmetical, then how can a simple statement to say that an axiom system is consistent (or is inconsistent)?
A fact that is easy to prove in the course of logic is that if an axiom system is inconsistent then every statement is provable from it. Therefore, to say that a system is consistent enough to say that there is any statement that is unprovable.
encoding Back to Kurt. Recall that, according to her, provable statements (for the specific system of axioms that have given us) are exactly those which code is a prime number that can be written as a sum or difference of three consecutive primes.
Thus, according to Kurt encoding, a way of affirming that the system is consistent is to say:
CONS (1): "There is a cousin who is addition or subtraction of three consecutive primes."
(The statement, as always, be translated into formal language.) Interestingly, according to the classification of David, this statement CONS says nothing about the consistency or inconsistency of the system.
Now, let's take a specific set P either. If the system is inconsistent then, as mentioned above, both P and not-P are both provable. If the system is consistent, however, at least one of the two statements (P or its negation) is not provable. That is, given the specific set P, the system is consistent if and only if P or not-P (at least one of the two) is not provable.
Suppose, as codified in Kurt, denial of the statement is the statement code 29 code 101. Therefore, the following statement also shows a way to assert that the system is consistent:
CONS (2): 29 or 101, at least one of them, can not be written as a sum or difference of three consecutive primes. "
Moreover, as we know, the system allows show all finitary statements true. In particular, can demonstrate the statement "not-(1 + 1 = 1)." Therefore, the system is consistent if and only if the statement "1 + 1 = 1" is not provable. Suppose that this last statement applies, according to Kurt, the number 2. We then have another way of stating that the system is consistent:
CONS (3): "2 can not be written as a sum or difference of three consecutive primes."
Take the three statements:
CONS (1): "There is a cousin who is addition or subtraction of three consecutive primes."
CONS (2): 29 or 101, at least one of them, can not be written as a sum or difference of three consecutive primes. "
CONS (3): "2 can not be written as a sum or difference of three consecutive primes."
Are these three statements are equivalent? This means: taking either of them as a premise, we can deduce the other two? Syntactically, the answer is no. The second or third set to deduce the first, but the former does not follow any of the otos two. In addition, the second does not follow the third, or vice versa.
What happens from the semantic point of view? In this case the universe of discourse should be the natural numbers (since we are thinking in terms of codes) and it is clear that, semantically, at arithmetic statements are not equivalent.
However, all three statements are equivalent to: "The system is consistent" and if all three amount to the same claim then they are equivalent to each other! I ask again: how to solve this paradox?
The answer is the same that we gave when speaking of self-reference. CONS (1), CONS (2) and CONS (3) are arithmetical. The interpretation of its meaning as referring the consistency of an axiom system is purely extramatemática (supra-arithmetic might say) and depends on the choice of a specific encoding (election that is alien to Arithmetic).
Denial of CONS (1) is (to be translated into formal language): "Every prime is addition or subtraction of three consecutive primes." Gödel's second theorem says that the system of axioms is consistent with that statement. Consistency, as we said, is a syntactic concept and interpretation of non-CONS (1) as "The system is consistent" is in another language level (beyond arithmetic) as does not conflict with the notion consistency. That is, neither more nor less, the resolution of the paradox posed at the beginning: the apparent paradox arises from the fact of mixing syntactic concepts such as consistency with concepts (let me word) above-semantic, the interpretation of CONS (1 ) in terms of consistency of the system. (1)
A little metaphor: usually red means "danger" and green means "security." To get the purple mix (or something). "Purple represents then a mix of danger and safety? "Purple = seguligro? "Purple = peliguro? It is obvious that the question meaningless and only comes to the same level of color mixing (physical appearance or syntax) with the interpretation that, culturally, we give to those colors (look above-chromatic). Similarly, Gödel's paradox arises from mixing concepts at different levels of analysis.
continued ...
Note:
(1) Further along the supra-semantic and see CONS (1) as the claim that the system is consistent, then CONS (1 ) is a statement true but unprovable in the system. As the system can prove all true statements finitary then we get the conclusion that the fact that no consistent system can be verified mechanically in a finite number of steps. This demonstrates the impossibility of reaching any of the requirements of the Hilbert Program, resourceful and have a complete system of arithmetic whose consistency can be verified algorithmically.
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