Our method is based on the same principles which we have assumed in a paper about the foundations of metamathematics which we published in collaboration with J. Herzberg some years ago. Nevertheless we have introduced some important modifications which enable us to use a very restricted preliminary language and to avoid any artificial construction.
The problem of a restricted language in mathematics appears to be a very important one, as our ordinary language is really ambiguous and obscure. On the other hand there is the problem of restricting the intuitive logic to be used in our construction. We shall see that for this purpose we need only substitutions and modus ponens.
If we avoided any abbreviation, we could reduce our preliminary language to propositions of the following forms:
X is a theorem.
If P, then Q.
But we have to do with very long symbolic expressions and as there is not yet any machine to build up such expressions we must use abbreviations. For this purpose we shall use propositions of the following forms in the preliminary language:
X is an expression.
X is an abbreviation of Y.
Any proposition which does not belong to one of our four forms will be excluded from our construction.