( A Part 8)
all true arithmetical sentence is provable
The main intention of this series of posts was clear some "frequently asked questions" relating to the theorems Gödel. For example: How can an axiomatic system is consistent with the statement that says that this system is not consistent? Can it be demonstrably a false statement? Etc.
As we saw in Gödel's theorem is three levels of language analysis. On one side is the syntactic level, where language symbols are manipulated without regard to its possible meaning (at this level are set out and prove Gödel's theorem).
We then semantic level in which the statements are understood as referring to a certain universe of discourse. This universe may be the natural numbers or other larger universe. This is the level at which mathematicians move in their daily work and is, in general, the level at which we are most comfortable.
At the third level (in the chapter Previous di-called supra-arithmetical ), thanks to a coding arbitrarily defined, some statements can be understood as referring to themselves metamathematics properties or other statements. It is only in this third level of analysis which appears self-reference or reference to the consistency of a system of axioms.
Most of the confusion concerning Gödel's theorems arise from improperly mixing concepts belonging to different levels. It is indeed a mixed error because, for example, as we saw in the previous chapter, there are statements that are equivalent in level, but not equivalent at a different level.
As I said in a previous chapter, self-reference is not essential to the proof of Gödel's theorems. The interpretation of a statement as saying "I am not provable" only helps to make more acceptable, or credible evidence of Gödel, but not an essential part of it. Why is there so much emphasis on self-reference, then? For mathematicians, believe it or not, are cold calculating machines, but human beings, and demonstrations (especially those made for the mathematician to mathematician B to be convinced that what he is saying is true) should not only be correct but also convincing.
I can not conclude this series of posts without referring to another common mistake in relation to Gödel's First Theorem. This error appears, for example, in the novel Uncle Petros and Goldbach Conjecture . Uncle Petros, the protagonist of the novel, is a very talented mathematician, a relatively young age, is obsessed with the idea of \u200b\u200bproving the Goldbach Conjecture.
few years after starting his company, Petros finds out the news of the demonstration of Gödel and the question arises whether the Goldbach Conjecture will not be an undecidable statement. Petros believes have the talent to find a proof of the conjecture, if this show exists. But with the possibility that perhaps the conjecture is true, but unprovable, decides to abandon the attempt to try and, in fact, abandon all mathematical research.
However, in previous chapters we have always spoken of "demonstrable with respect to some set of axioms" or "undecidable with respect to such a system of axioms." But there are undecidable statements in an absolute sense? Are there truths unprovable? Why do I say "system axioms" and not "set of axioms?
Answer the second question first. When it comes to demonstrations, consistency, etc. we must not only take into account the axioms of the theory in question, but we must also take into account the underlying logic and rules of inference. I speak of "system axioms" as equivalent to "set of axioms underlying logic + + rules of inference."
The underlying logic consists of language statements that are "universally valid." That is, statements which are true whatever the interpretation given to the symbols of language. Note that this definition is semantic. An example is "P or not-P "where P is a statement either. We speak of" rationale "for these statements can be part (and, indeed, form part) for any reason.
If language restrictions we discussed in an earlier chapter, then it is possible to give a recursive system of axioms that allow deduction of all valid sentences expressible in that language. This fact was proved by Gödel in 1929 (his incompleteness theorems are from 1931), we enables a definition syntactic of the notion of universally valid statement "are all those which follow, modus ponens through, system of axioms that Gödel did in 1929 (or any other equivalent to it).
languages \u200b\u200bthat respect the above restrictions are called first-order languages \u200b\u200b. is said then that the first-order logic is formalizable or is recursively axiomatizable. As the first-order logic is recursively axiomatizable then there are chances that a theory in a language and to comply with conditions of Hilbert's program (the arithmetic does not comply, that Gödel proved, but if they can meet other theories).
increases now the power of language through the gimmick of adding variables relating to statements or formulas (a formula is an expression in which there are "free variables" that can be replaced by random numbers, such as " x is even ").
We now have a language second order or higher order . In this type of language we can say, for example (in a first-order language can not express the statement that follows):
" a = b if and only if for every formula P (x ), P ( to ) is equivalent P (b ) "
languages \u200b\u200bin second order we also have the concept of valid sentences (and which are the" underlying logic "of the theories expressed in these languages). However, can prove that it is impossible to give a recursive set of axioms to prove all valid sentences of second order.
The second-order logic, then there is no syntax definition for the underlying logic, and hence the Hilbert program is unworkable level theories expressed in such languages.
All demonstration in a language of second order is non-finitary in the sense that the underlying logic is not definable syntactically (and not because it has an infinite number of steps.) The inference rule used in the second-order logic, nor is finite, is as follows: "Q follows from P if for any interpretation of the symbols for which P is true is that Q is also true" .
However, if the language we gave earlier to the Arithmetic we add variables to the statements and formulas (and transformed it into a language of second order) then it is possible to give a finite set of axioms (namely Axioms Peano) such that every statement real arithmetic is provable from them .
Again, all true arithmetical sentence is provable from the Peano axioms, if we admit a second-order logic. (The theory is undecidable, but are not arithmetical). In particular, if Goldbach's conjecture is true then it is provable.
Although impossible to predict how difficult it is to find a show of it, nor will it be possible to give an algorithm to verify its correctness. So did Uncle Petros abandobar badly in your search.
ends this series of entries here. I hope to have resolved some doubts, but mainly I hope to have raised doubts new, better and stronger.
( A Part 8)