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Register a new userUnprovability in Mathematics: A First Course on Ordinal Analysis
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Anton Freund
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These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskals theorem for binary trees is unprovable in conservative extensions of Peano arithmetic (note that much stronger results of this type are due to Harvey Friedman). Concerning prerequisites, we assume a solid introduction to mathematical logic but no specialized knowledge of proof theory. The material in these notes is intended for 14 lectures and 7 exercise sessions of 90 minutes each.
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For every partial combinatory algebra (pca), we define a hierarchy of extensionality relations using ordinals. We investigate the closure ordinals of pcas, i.e. the smallest ordinals where these relations become equal. We show that the closure ordinal of Kleenes first model is $omega_1^textit{CK}$ and that the closure ordinal of Kleenes second model is $omega_1$. We calculate the exact complexities of the extensionality relations in Kleenes first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pcas.
Ramseys theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramseys theorem for pairs due to Erdos and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type eta, where eta is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramseys theorem for pairs. This Erdos-Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.
This is a set of 288 questions written for a Moore-style course in Mathematical Logic. I have used these (or some variation) four times in a beginning graduate course. Topics covered are: propositional logic axioms of ZFC wellorderings and equivalents of AC ordinal and cardinal arithmetic first order logic, and the compactness theorem Lowenheim-Skolem theorems Turing machines, Churchs Thesis completeness theorem and first incompleteness theorem undecidable theories second incompleteness theorem
Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system $mathsf{RCA}_0 + mathsf{I}Sigma_2$. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bezout and GCD domains.
We present three ordinal notation systems representing ordinals below $varepsilon_0$ in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.
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