From the point of view of formal proofs the system T has no "semantics", otherwise, the sense of the symbols used in it to us is indifferent. The formal proof is only some long chain of lines in which every line is an axiom T, a general-logical axiom, or is received from the last lines by application of one of the allowed rules of transition. We designated, say, one of operations of language of arithmetics a symbol * because it corresponds to our understanding of multiplication; but from the point of view of formal system T * — only a symbol which means nothing. Instead of it there could be any other symbol, say, of %, and all proofs would remain in force; simply if we wanted to define sense of axioms or the theorems proved by us, we should understand % as "multiplication".
Let T — "suitable" formal system, also we will assume again that T is correct. Then we can construct the concrete statement of G (called by "the statement"), possessing the following property: G is true, but is unprovable in T.
where - any formula of the theory of natural numbers. The ninth axiom is called as the principle of mathematical induction. Axioms 1-2 provide obvious properties of equality, axioms 5-8 specify properties of operations of addition and multiplication.
. Language of the theory of the first order. Let's consider some alphabet of the theory the Set of words of this alphabet is called as a set of expressions of the theory Couple consisting of the alphabet and a set of expressions call theory language.
Incompleteness of system T is approved as result only in the third version, but it is easy to see that it follows at once from the conclusion and in the first two versions. We conclude in them that there is some true, but unprovable statement. Such statement of T does not prove, but also to disprove it — to prove its denial — it cannot since its denial is false, and T (in the first two versions of the theorem) is correct and proves only true statements. Therefore T cannot neither prove, nor disprove such statement of G and, therefore, T is incomplete.
There are formal systems which prove only true statements. Systems, in which all axioms — true statements are that (it is possible to prove that then all rules of transition between axioms keep the validity). Such formal systems are called correct.
Self-applicability recognition problem. It is the second problem which positive solution is not found still. Its essence consists in the following. The program of the car of Turing can be coded any certain code. On a tape of the car it is possible to represent its own code which is written down in the alphabet of the car. Here as well as in case of the usual program two cases are possible: