A formal system is a whole of formulas, or formal expressions, which one can interpret like names, sentences, or in any other way. They are fundamental whole for logic and mathematics.

Examples

• the whole of numbers, entireties, rational, algebraic, can be defined like formal systems, but not the units which contain all the transcendent numbers, real or complex.

• the nomenclature of the organic chemistry is a formal system.

• a Théorie is a whole of sentences, or proposals, and is thus a formal system.

General problems

The general theories of the formal systems were conceived by logicians especially to study the theories. From this point of view one can regard them as general métathéories, theories of all the theories.

The three fundamental problems for the formal system theories are: - How is whole of formulas defined? - How does one interpret the formulas which speak about the formulas and the whole of formulas? - How does one prove truths in connection with the whole of formulas?

The formal point of view introduces a limitation. One worries little about the significance of the words or the symbols. The theories are not regarded there as windows on the real-world. They are opaque. They contain only assemblies of words and one is concerned initially with their forms, not of their significances. From where the formal name from point of view.

Axiomatic theories

A formal system is often built by giving itself a whole of Axiome S and while reasoning starting from these axioms using the usual Logique. For example, the axiomatic Théorie of the units is a formal system. Let us recall initially that an axiom is a not shown proposal which is used as starting point with a reasoning (for example “by two points it passes one and only one line” is an axiom of the Euclidean geometry). A Théorème is a proposal deduced starting from the axioms, in a finished number of stages, with the rules of logic. If the rules of deduction are valid, a theorem is true provided that the axioms are true.

The truth of the axioms or the formulas is defined relative with a model, a possible universe, in which the formulas are interpreted.

Enumerability of the axiomatic theories

The basic formal systems are units énumérables. Intuitively, it is all the units for which one can give a mechanical process of enumeration of all their elements.

An axiomatic theory is always énumérable, for the following reasons.

• the list, finished or infinite, of its axioms, is always Décidable, because one wants to know precisely what is and what is not an axiom.

• the formal methods impose that the rules of deduction are mechanical, that they can be applied blindly by a machine. The whole of the formal evidences is thus always décidable. If one presents a proof formalized to a suitably programmed computer, he answers so yes or not the proof is correct, so yes or not it starts with axioms and complies with the rules of deduction.

The theory, i.e. the whole of the theorems, or formulas provable starting from the axioms, is énumérable, because one can define an order on the whole of all the finished lists of formulas. That is to say a formula F which one wants to know if it is a theorem. The computer examines each finished list of formulas one by one and decides so yes or not it is an formal evidence. If so, then the last formula of the list is a theorem. If this formula is F then the computer stops and answers that F is a theorem. In the other cases, the computer examines the following finished list.

If F is really a theorem, a computer thus programmed will always find the answer, because it examines all the possible formal evidences. But it will put much time, too much so that this method is really effective for us simple mortals.

If F is not a theorem, the computer never stops, it examines without stop of new lists, it finds new evidence, but it will never find proof of F, since F is not a theorem. An axiomatic theory is thus always énumérable but it is not sure a priori that it is décidable.

“A good” formal system S must be coherent.

S is coherent if it does not make it possible to show a contradiction formally (: if there does not exist proposal has such as S shows has and not A)

If it is not coherent and the principle of the reasoning by the absurdity is accepted among the rules of logical deduction then all the formulas are provable. A system which proves all and its opposite does not prove anything the whole.

Kurt Gödel showed at the beginning of the century that any formal system so much is not very complex contained true but nondemonstrable proposals (see Théorème of incomplétude of Gödel).

See also

• the definition of Smullyan of the units énumérables

Sources

• Raymond Smullyan, Theory off formal systems

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