To tell that some statement is demonstrable in T — means to tell that there is some formal proof which brings to it. Provability — syntactic property, but not semantic. On the other hand, to tell that some statement is true — means, to tell that if we interpret it according to usual interpretation of symbols T (i.e. * we will understand as "multiplication", a symbol 0 — as number 0,. ), we receive the true statement about natural numbers.
Gedel's theorem of incompleteness. In any consistent formal system containing arithmetics minimum, and, therefore, and in the theory of natural numbers, there will be formally unsoluble judgment, that is such closed formula that, are not removed in system.
Axiomatic theories share on formal and informal. Informal axiomatic theories are filled the theorist – the multiple contents, the concept of deductibility of them quite vaguely and substantially relies on common sense.
For any theories of the first order the theorem of deduction proved by us in calculation of statements demands change. In an original form, and any restrictions on the subject variables entering in it was not imposed. For justice of the theorem of deduction for any theories of the first order it is necessary to change it as follows.
First, the first theorem of Gödel's incompleteness is used in the proof of the second theorem of Gödel's incompleteness which proves that "suitable" (in slightly another, but similar with described above, sense the formal system T cannot prove own consistency if it is a konsistentna (if it is a nekonsistentna, it can prove anything, including own consistency as paradoxically it sounds). I will not go into details, but I will notice only that in the course of the proof of the second theorem of incompleteness it is necessary to show that the proof of the first theorem of incompleteness can be formalized in system T. In other words, it is not simple "if T konsistent, it is incomplete" (the third version of the first theorem of incompleteness, see, but also this statement (more precisely, anat its arithmetic it is possible to prove in the system T. But while it is possible to formalize such concepts as "formal system", "consistency" and "completeness" "in" system T, it appears that the concept "validity" cannot be formalized in T in principle. Therefore the first and second versions of the theorem of Gödel though they are also simpler for the proof, cannot be used for the proof of the second theorem of Gödel.
Let T — "suitable" formal system, also we will assume that T konsistentn. Then T is not full system, i.e. there is a statement of G such that T cannot it neither prove, nor disprove; moreover, we can construct such concrete G (called by "the statement").
That in the strongest and general wording Gödel's theorem does not impose on T any essential semantic conditions, and the conclusion it too quite sintaksichno — it it is very important to understand. Important not only and not so much because sometimes we want to apply Gödel's theorem to incorrect systems, though it too is right. Important generally for the following two reasons.
Theories of the first order are distinguished from mathematical theories. These theories do not allow predicates which have other predicates and functions as arguments in the statement. Besides, kvantorny operations on predicates and functions are not allowed. Theories of the first order are called still as elementary theories.
The formal system is called jellied if it cannot prove at the same time any statement and its denial, i.e. prove a contradiction. Not jellied formal system — it is bad and almost useless since it is possible to show easily that from the proof of a contradiction it is possible to obtain the evidence anything. Not jellied formal system proves in general any statement so anything interesting in it is not present.
Set of all words in some alphabet together with any final system of admissible substitutions is called as associative calculation. For a task of associative calculation it is enough to set the corresponding alphabet and system of substitutions.