(Part 2 A - Part 4 A )
Demos
- Professor, when we self-reference?
- Patience, and almost there.
In the previous chapter we said that the program proposed to Hilbert axioms for arithmetic and axioms (and rules of inference, which are those that tell us what conclusions can be obtained from certain assumptions) should be chosen so that the correction of any show based on them could be verified (from the meta-level) mechanically in a finite number of steps (ie, it should be possible to program a computer to check whether a proof is valid or not.)
Specifically, the demonstrations that included the Hilbert program as valid should be translatable into a finite sequence of statements such that each of these, or it was an axiom, or it could be inferred from statements previously located in the succession by applying certain specific inference rules. (1)
This idea imposes three conditions for the formalization of arithmetic. The first is that their statements must be translated into a language with well-defined symbols (a prerequisite to have an algorithm that works on these statements-a meta-level-).
For the purposes of this series of posts, the language we use consist of the symbol "+" and ".", The constant 1, which add parentheses and symbols for logical operations. The language will also variables, x, y, z, ... may represent only natural numbers (never express functions, sets or other objects (2) ).
look
metamathematics that works only at the syntactic level, so the expression
(1 + 1). (1 + 1) = 1 + 1 + 1 + 1
shall, at the meta-level, different expression:
1 + 1 + 1 + 1 = (1 + 1). (1 + 1)
because, although both have the same symbols, they are written in different order.
Assume, for meta-treatment, all statements have been translated into this formal language. Assume also that we have a sequence of statements and you want to write a program to test whether this sequence is, or is not a valid demonstration. The "take" then a statement of the sequence and must verify whether it is or not an axiom.
The second is, then, that there is an algorithm to verify in a finite number of steps if a statement is or not an axiom.
Continuing the process that should continue this program, if a statement is not an axiom, the program must be able to check whether the statement can be deduced from previous statements in succession. The third characteristic is then the ratio "Q follows from the hypotheses H1, H2, H3 ,..." must be verified algorithmically.
In fact, we reduce ourselves to take a single inference rule, called Rule of Modus Ponens, which says that P and P ---> Q Q. It follows (The rule must be understood at the syntactic level, without appeal to possible meanings.)
say that a property is recursive if algorithmically verifiable. We can say that the system of axioms and inference rules must be both resourceful.
continued ...
Notes:
(1) This is the meta-verification process of proposing demostraciopnes Hilbert Program. Of course, there is the process by which mathematicians find these demonstrations.
(2) Every theory has two types of axioms: its specific axioms and the logical axioms , that are general and common to all theories. These are statements that are worth whatever the universe of discourse under consideration (eg, "To everything x, x = x ). If we respect restrictions on the use of variables then it is possible to give an axiom system that respects the conditions for Hilbert and to deduce all these universally valid statements (this was proved by Gödel in 1929). If we admit variables representing functions, sets, etc. Hilbert then the program would be unworkable at the level of the same underlying logic.
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