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Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets

Tim Button

Bulletin of Symbolic Logic 27 (4):436-460 (2021)

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A Taxonomy for Set-Theoretic Potentialism.Davide Sutto - 2024 - Philosophia Mathematica:1-28.details Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. (...) Download Export citation Bookmark
The purely iterative conception of set.Ansten Klev - 2024 - Philosophia Mathematica 32 (3):358-378.details According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive set theory. I will explain Aczel’s model construction in a way that presupposes no previous familiarity with the theories on (...) which it is based. (shrink) Download Export citation Bookmark
Wand/Set Theories: A realization of Conway's mathematicians' liberation movement, with an application to Church's set theory with a universal set.Tim Button - forthcoming - Journal of Symbolic Logic.details Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) Download Export citation Bookmark
Symmetric relations, symmetric theories, and Pythagrapheanism.Tim Button - 2022 - Philosophy and Phenomenological Research (3):583-612.details It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph. Download Export citation Bookmark 1 citation
Against Cumulative Type Theory.Tim Button & Robert Trueman - 2022 - Review of Symbolic Logic 15 (4):907-49.details Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions imposed by (...) Download Export citation Bookmark 10 citations

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