Patterns of resemblance and Bachmann-Howard fixed points

Timothy Carlson’s patterns of resemblance employ the notion of $$\Sigma _1$$ Σ 1 -elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patterns to dilators leads to an equivalence with $$\Pi ^1_1$$ Π 1 1 -comprehension (Question 27 of A. Montalbán’s “Open questions in reverse mathematics”, Bull. Symb. Log. 17(3)2011, 431-454). In the present paper we prove this conjecture. The crucial direction of the equivalence (towards $$\Pi ^1_1$$ Π 1 1 -comprehension) is reduced to a previous result of the author, which is concerned with relativizations of the Bachmann-Howard ordinal.

Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Reverse mathematics of rings

<p>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 RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>

Reverse mathematics of rings

<p>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 RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>

Countable sets versus sets that are countable in reverse mathematics

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic L 2 . A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets must be represented by sequences here, because the higher-order definition of ‘countable set’ involving injections/bijections to N cannot be directly expressed in L 2 . Working in Kohlenbach’s higher-order RM, we investigate various central theorems, e.g. those due to König, Ramsey, Bolzano, Weierstrass, and Borel, in their (often original) formulation involving the definition of ‘countable set’ based on injections/bijections to N. This study turns out to be closely related to the logical properties of the uncountably of R, recently developed by the author and Dag Normann. Now, ‘being countable’ can be expressed by the existence of an injection to N (Kunen) or the existence of a bijection to N (Hrbacek–Jech). The former (and not the latter) choice yields ‘explosive’ theorems, i.e. relatively weak statements that become much stronger when combined with discontinuous functionals, even up to Π 2 1 - CA 0 . Nonetheless, replacing ‘sequence’ by ‘countable set’ seriously reduces the first-order strength of these theorems, whatever the notion of ‘set’ used. Finally, we obtain ‘splittings’ involving e.g. lemmas by König and theorems from the RM zoo, showing that the latter are ‘a lot more tame’ when formulated with countable sets.