calculus solution document

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The Practice of Finitism: EpsilonConsistency Proofs in Hilberts ProgramCalculusandRichard Zach (rzach@ucalgary.ca)University of CalgaryAbstract. After a brief flirtation with logicism around 1917, David Hilbert proposed hisown program in the foundations of mathematics in 1920 and developed it, in concert withcollaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The twotechnical pillars of the project were the development of axiomatic systems for ever strongerand more comprehensive areas of mathematics, and finitistic proofs of consistency of thesesystems. Early advances in these areas were made by Hilbert (and Bernays) in a series oflecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermanns dissertation of 1924. The main innovation was the invention of the -calculus, onwhich Hilberts axiom systems were based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the development of the simultaneousdevelopment of logic and mathematics through the -notation and provides an analysis ofAckermanns consistency proofs for primitive recursive arithmetic and for the first comprehensive mathematical system, the latter using the substitution method. It is striking that theseproofs use transfinite induction not dissimilar to that used in Gentzens later consistencyproof as well as non-primitive recursive definitions, and that these met

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