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An alternative to relative proofs of consistency was proposed by Hilbert. He sought to construct “absolute” proofs, by which the consistency of systems could be established without assuming the consistency of some other system. We must briefly explain this approach as a further preparation for understanding Go¨del’s achievement. The first step in the construction of an absolute proof, as Hilbert conceived the matter, is the complete formalization of a deductive system. This involves draining the expressions occurring within the system of all meaning: they are to be regarded simply as empty signs.
Gʚp bʚp Іgʚb gp ϭ 0 bp ϭ 0 gb ϭ 0 Symbolic logic was invented in the middle of the 19th century by the English mathematician George Boole. In this illustration a syllogism is translated into his notation in two different ways. ” Thus ‘g ʚ p’ says that the class of gentlemen is included in the class of polite persons. In the lower group of formulas two letters together mean the class of things having both characteristics. For example, ‘bp’ means the class of individuals who are bankers and polite; and the equation ‘bp ϭ 0’ says that this class has no members.
This deals with relations between statements containing such “quantifying” particles as ‘all’, ‘some’, and their synonyms. We shall exhibit one elementary theorem of logic and one rule of inference, each of which is a necessary but silent partner in the demonstration. The Systematic Codification of Formal Logic 39 Look at line 5 of the proof. Where does it come from? The answer is, from the logical principle (or necessary truth): ‘Either p or non-p’, where ‘p’ is called a sentential variable. But how do we get line 5 from this theorem?