000			a
999			_c30147 _d30147
008			210128b xxu||||| |||| 00| 0 eng d
020			_a9780691196411
082			_a511.3 _bSTI
100			_aStillwell, John
245			_aReverse mathematics : proofs from the inside out
260			_bPrinceton University Press _c2018 _aPrinceton
300			_axiii, 182 p. _bill. _c24 cm
365			_b18.95 _cUSD _d76.50
490			_aProQuest Ebook Central
504			_aIncludes bibliographical references and index.
520			_aThis book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.
650			_aReverse mathematics
650			_aArithmetization of Computation
650			_aComputability
650			_aHillbert's Axioms
650			_aBolzano - Weierstrass theorem
650			_aCantor set
650			_aComprehension arithmetical
650			_aExtreme value theorem
650			_aRecursive comprehension
650			_aSet existence axiom
942			_2ddc _cBK