8. New Version of the Consistency Proof for Elementary Number Theory

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This chapter presents a new version of the consistency proof and emphasis on developing the fundamental ideas and on making every single step of the proof as lucid as possible. The formal representation of the forms of inference that is directly based, as it was on mathematical practice, with the characteristic concept of the “sequent,” proves quite suitable for metamathematical investigations; it is better suited to most purposes than the methods of representation generally customary till date. Nevertheless, it cannot be said that the “most natural” logical calculus, simply because it corresponds most closely to real reasoning, is also the most suitable calculus for proof-theoretical investigations. For the consistency proof, in particular, a somewhat different version has proved to be even more suitable. An inference figure (the formal counterpart of an inference) consists of a line of inference, a lower sequent (written below the line), and upper sequents (one or more), written above the line. The lower sequent here stands for the conclusion of the inference that has been drawn from the premises represented by the upper sequents.