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By Bruno Poizat

Translated from the French, this booklet is an creation to first-order version idea. ranging from scratch, it quick reaches the necessities, particularly, the back-and-forth procedure and compactness, that are illustrated with examples taken from algebra. It additionally introduces common sense through the research of the types of mathematics, and it supplies entire yet available exposition of balance theory.

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On the other hand, in II we have a similar enumeration of all the sets of arguments for which those predicates are significant. The two together define the predicates in extension. In extenso then - assuming that we know what the terms are the predicates are uniquely defined by the primitive frame. In regard to the terms, however, the situation is entirely different. We have seen that concerning the tokens the primitive frame says nothing whatever. We can therefore take for those tokens any objects we please, and similarly we can take for operators any ways of combining these objects which have the requisite formal properties 1.

One of its achievements is that it enables us to think of a formal system as something very concrete without losing sight of abstractness, and so incidentally to show that we do not need to presuppose mystical entities of a logical or other idealistic kind in order to be formal. But it has also certain disadvantages, and the time has come when we should sign a declaration of independence from it also. IX METATHEORY In the foregoing we have been considering a formal system essentially as a body of elementary propositions.

There is, however, another reduction which has been accomplished in Example 8, viz. the elimination of variables. This matter is too complicated for us to go into here. However, it has been shown that Example 6 can be reduced, in a certain sense, to a sub-system of Example 8. I conjecture that practically any system can be formulated without variables by making appropriate additions to Example 8. 1 We can characterize a completely formal system as one in which all non-trivial recursive definitions necessary to define the system are made theoretically.

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