Trakhtenbrot's theorem

The set of costume bread, named after Boris costume bread, is a set of mathematical logic. It was proved in 1950 and states that the generally applicable in all finite models sentences of first-order predicate logic are not ordinal. This has consequences for the predicate second-order logic.

Formulations of the sentence

It is the symbol set with countably many constant symbols and countably infinite for each arity many function and relation symbols. Next is the set of all sets of first-order logic that are true in all finite structures. Then:

Abstract: The generally valid in the finite sets of first stage are not enumerable.

Next is the set of all sets, for which there is a structure in which they are met. Then:

Abstract: The satisfiable in the finite sets of first stage are not decidable.

Note to proof

The second formulation is attributed to the unsolvability of the halting problem, in which one exploits that can be described in finite models Turing machines. This then results in the holder first named, because the generally applicable in the finite sets are precisely the negations of not satisfiable in the finite sets.

Incompleteness of the second -order predicate logic

As an application, we show how from the set of costume bread incompleteness of the second -order predicate logic. It is the set of the sets of second -order logic that are valid in all models. Then:

This situation is called the incompleteness of second -order predicate logic. For a proof of a theorem of second- order predicate logic, which applies exactly in finite models, such as

That is, applies in all cases, when a function is and is injective, then is surjective. Now if enumerable, so you start an enumeration process and whenever this creates a statement of the form with a set of first-order predicate logic, give it out. In this way, all generally valid in the finite sets of first stage are enumerated, which contradicts the set of costume bread.