Subsystems of Second Order ArithmeticFoundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic. Additional results are presented in an appendix. |
Contents
lbN | 14 |
Foundational programs and the five basic systems | 43 |
Part B Models of Subsystems of | 52 |
Models of subsystems of Z2 | 54 |
ARITHMETIcAL CoMPREHENsIoN | 105 |
WEAK KoNIGs LEMMA | 127 |
ARITHMETIcAL TRANSFINITE RECURSION | 167 |
111 COMPREHENSION | 217 |
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Common terms and phrases
Abelian group ACAO ACAO proves algebraic closure analytic code analytic sets arithmetical comprehension arithmetical transfinite recursion assume ATRO Banach space Borel code Borel sets Cantor space chapter commutative ring compact metric space complete separable metric completes the proof consisting continuous function COROLLARY countable Abelian group countable field countable well ordering define descriptive set theory determinacy equivalent over RCAO find finite sequences first order following definitions following is provable free variables g x g HCL(X Hence hyperarithmetical i-model implies infinite isomorphism metric space nonempty open set ordinary mathematics parameters path primitive recursion proof of lemma proof of theorem provable in RCAO quantifiers Ramsey theorem real numbers Reverse Mathematics satisfies sentences separable Banach space separable metric space soundness theorem subset subsystems of Z2 transfinite induction tree Turing jump weak KOnig’s lemma WK L0 WKLO WO(X