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By assigning ordinal notations to proofs in classical arithmetic it is possible to show that each step in the simplification process making up the consistency proof, the complexity of proofs, as measured by the associated ordinal notation, successively decreases. Since the system of ordinal notations is well-ordered, it is not possible to have an infinite decreasing sequence. The reduction process described in Chapter 7 therefore must bring down the ordinal notations assigned to the successive transformed proofs, and must ultimately end with a proof whose ordinal notation is less than ω, i.e. a “simple” proof. Since “simple” proofs cannot be proofs of a contradiction, this establishes the consistency of classical arithmetic. |
