Freund, Anton Jonathan (2018) TypeTwo WellOrdering Principles, Admissible Sets, and Pi^1_1Comprehension. PhD thesis, University of Leeds.

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Abstract
This thesis introduces a wellordering principle of type two, which we call the BachmannHoward principle. The main result states that the BachmannHoward principle is equivalent to the existence of admissible sets and thus to Pi^1_1comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory. A typeone wellordering principle is a map T which transforms each wellorder X into another wellorder T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our BachmannHoward principle transforms each dilator T into a wellorder BH(T). The latter is a certain kind of fixedpoint: It comes with an "almost" monotone collapse theta:T[BH(T)]>BH(T) (we cannot expect full monotonicity, since the ordertype of T[X] may always exceed the ordertype of X). The BachmannHoward principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed". On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for settheoretic reverse mathematics; a formalization of dilators in weak set theories and secondorder arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schütte) and betacompleteness (Girard) in a settheoretic context; and a betaconsistency proof for KripkePlatek set theory. Independently of the BachmannHoward principle, the thesis contains a series of results connected to slow consistency (introduced by S.D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the ParisHarrington principle can only have extremely long proofs in certain fragments of arithmetic.
Item Type:  Thesis (PhD) 

Keywords:  wellordering principles, admissible sets, Pi^1_1comprehension, dilators, betaproofs, BachmannHoward ordinal, primitive recursive set theory, slow consistency, ParisHarrington principle, proof length 
Academic Units:  The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) 
Depositing User:  Mr. Anton Jonathan Freund 
Date Deposited:  07 Aug 2018 10:02 
Last Modified:  07 Aug 2018 10:02 
URI:  http://etheses.whiterose.ac.uk/id/eprint/20929 
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