Wednesday, March 2, 2011

Firs Pap Smear Get A Brazilian Wax?

in the demonstration of Gödel (Part 4) The self-referential

( A Part 3 - Part 5 A )

Dialectics True / Demonstrable

intuitionist school, that was opposed to the Hilbert says, as we said, that mathematical objects are constructed by humans and are not pre-existing construction. For them, the math is created, not discovered and thus, for example, a number that has never been calculated simply does not exist. Therefore intuitionists sense to deny any claims that talk of wholes infnitas as necessarily talking about nonexistent objects.

Hilbert, in his program, proposes a more moderate approach. On the one hand speaks of claims Hilbert finitary (or finitarias , according to some translations), which are those whose truth or falsehood can be mechanically verficada a finite number of steps (for example, "12 + 3 = 15" "22 is not prime"). Of these statements is true, no problems, they are true or false, as the case [the intuition would agree with this.]

Hilbert admits secondly, that the statements refer to infinite totalities are problematic and it is not easy to attribute a truth value. But unlike the intuitionist says that the attribution should be done. "Understanding the infinite," says Hilbert, "is a challenge to the human spirit", a challenge that we face and overcome.

Hilbert makes the following comparison: in mathematics, on the one hand we have the sums of a finite amount of numbers, which can be calculated without problem. But on the other hand, we also have the series, which are infinite sums. The series can become very problematic, however, thanks to the notion of limits, the Analysis has been able to give a clear and precise meaning. Similarly, says Hilbert, Logic has the challenge of giving a clear and precise meaning to statements not finitary.

This effect would be through the notion of provability . Given a set of axioms, we say that a statement P is provable from them if there is a demonstration (based on this system of axioms) whose last statement is P. [That is, if there is a show that ends with the statement P. The meta-definition demonstration we have seen in the previous chapter.]

Program Hilbert proposed, therefore, give axioms for arithmetic so that, for starters, all finitary statement is demonstrably true (it's the least we can ask for) and that, moreover, for any statement P, either P or its negation is provable.

Each statement "demonstrably" is "true." That is, the Hilbert program sought a synthesis between the concept of provability and truth. Since the concept of provability is syntactic (mechanically verifiable in a finite number of steps), we get a notion of truth "safe" and free from paradoxes.

Unfortunately for Hilbert, Gödel's first theorem proves that this is impossible: in Arithmetic "truth" and "demonstrations" are not equivalent. Whatever the choice axioms (provided they meet the conditions for Hilbert metamathematics) will always be some set P such that both he and his denial not demosrables (as P would be out of the syntactic definition of "truth").

In his book The Emperor's New Mind , Roger Penrose says he does not understand "contempt" Hilbert's followers feel about the notion of mathematical truth, and that support, according to Penrose, the possibility that has set that are not true or false.

In fact, Penrose is wrong to make this comment. It is true that if the assertion P is not provable, nor its negation is provable, then for the Hilbert program P would not be true or false. But in reality, the idea was to Hilbert axioms so that this situation never happened. Any statement, through the meta-definition of provability, it should be, ultimately, true or false. Was Gödel who spoiled the party to prove that there would always be undecidable statements.

Before ending a little recap what we've seen before here. The Hilbert program meant to give an axiom system for arithmetic that meets these conditions:

1. The system should be recursive (was due to mechanically verify a finite number of steps if a statement is or not an axiom).
2. The system should be consistent (there should be a statement P such that it and its negation be simultaneously demonstrable, the inference rule is modus ponens ).
3. all finitary statement should be demonstrably true (this is the condition which is sometimes expressed as "contains enough arithmetic.")

4. For every statement P, either P or its negation must be demonstrable.
5. had to be verified mechanically in a finite number of steps that the system is consistent.

As already mentioned, Gödel's theorems show that if the first three conditions are met then the last two fail. From the next chapter, we discuss the proof of these theorems and arrive, finally, the discussion of self-reference.

( A Part 3 - Part 5 A )

0 comments:

Post a Comment