Switch to: References

Citations of:

A Second Glance at Non-restrictiveness

B. Lowe

Philosophia Mathematica 11 (3):323-331 (2003)

Add citations

You must login to add citations.

Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.details In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink) Download Export citation Bookmark 12 citations
Restrictiveness relative to notions of interpretation.Luca Incurvati & Benedikt Löwe - 2016 - Review of Symbolic Logic 9 (2): 238-250.details Maddy gave a semi-formal account of restrictiveness by defining a formal notion based on a class of interpretations and explaining how to handle false positives and false negatives. Recently, Hamkins pointed out some structural issues with Maddy's definition. We look at Maddy's formal definitions from the point of view of an abstract interpretation relation. We consider various candidates for this interpretation relation, including one that is close to Maddy's original notion, but fixes the issues raised by Hamkins. Our work brings (...) Download Export citation Bookmark 7 citations
(1 other version)Are Large Cardinal Axioms Restrictive?Neil Barton - 2023 - Philosophia Mathematica 31 (3):372-407.details The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...) Download Export citation Bookmark 1 citation
Long Borel hierarchies.Arnold W. Miller - 2008 - Mathematical Logic Quarterly 54 (3):307-322.details We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω2. This implies that ω1 has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω2. We also show that assuming a large cardinal hypothesis there are models of ZF in which (...) Download Export citation Bookmark 2 citations

1