A note on the consistency operator

Walsh, James

doi:10.1090/proc/14948

Cited by 4 publications

(10 citation statements)

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“…We will explore the analogy between the recursion-theoretic and proof-theoretic well-ordering phenomena throughout this paper. On the one hand, we will describe proof-theoretic theorems from [22,23,24,33] whose statements and proofs were inspired by this recursion-theoretic research. On the other hand, we will also describe purely recursion-theoretic results from [19] that were inspired by the aforementioned proof-theoretic theorems.…”

Section: Turing Degree Theorymentioning

confidence: 99%

“…In [33] some limitative results on the scope of this approach are established. In particular, it is shown that the assumption that g is recursive is necessary in the statement of Theorem 8.6.…”

Section: Analogues Of Martin's Conjecturementioning

confidence: 99%

“…In fact, more can be said. Indeed, in [33] it is shown that sufficiently constrained recursive monotone functions must coincide with the consistency operator in the limit within the ultrafilter of true sentences. Note that "in the limit" plays a role analogous to that which "a.e.…”

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confidence: 99%

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On the Hierarchy of Natural Theories

Walsh

2021

Preprint

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It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research.

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Section: Turing Degree Theorymentioning

confidence: 99%

Section: Analogues Of Martin's Conjecturementioning

confidence: 99%

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On the Hierarchy of Natural Theories

Walsh

2021

Preprint

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“…In [4,6] an approach to this problem was proposed. The approach in question was inspired by Martin's approach to an analogous question in recursion theory: Why are the natural Turing degrees well-ordered by Turing reducibility?…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, Martin's Conjecture says that the only definable degree-invariant functions, up to Martin-equivalence, are the constant functions, the identity function, and the iterates of the Turing jump. 1 The optimistic proposal in [4,6] was that an analogue of Martin's Conjecture holds for axiomatic theories. Let's fix a sound recursively extension T of elementary 1 Of course, the notion of "definable" is left vague in what I have written.…”

Section: Introductionmentioning

confidence: 99%

Evitable iterates of the consistency operator

Walsh¹

2022

Preprint

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Let's fix a reasonable subsystem T of arithmetic; why are natural extensions of T pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of T . According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate Con α T of the consistency operator in the limit.In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as Con T in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as Con T in the limit nor as strong as Con 2 T in the limit. In fact, for every α, we produce a function that is cofinally as strong as Con α T yet cofinally as weak as Con T .

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Evitable iterates of the consistency operator

Walsh

2023

COM

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Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin’s Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem T of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of T. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate Con T α of the consistency operator “in the limit” within the ultrafilter of sentences that are true in the standard model. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as Con T in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as Con T in the limit nor as strong as Con T 2 in the limit. In fact, for every α, we produce a function that is cofinally equivalent to Con T α yet cofinally equivalent to Con T .

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A note on the consistency operator

Cited by 4 publications

References 4 publications

On the Hierarchy of Natural Theories

On the Hierarchy of Natural Theories

Evitable iterates of the consistency operator

Evitable iterates of the consistency operator

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