author={Thomas E. Forster},
  journal={The Review of Symbolic Logic},
  pages={97 - 110}
  • T. Forster
  • Published 1 June 2008
  • Psychology
  • The Review of Symbolic Logic
The two expressions ‘The cumulative hierarchy’ and ‘The iterative conception of sets’ are usually taken to be synonymous. However, the second is more general than the first, in that there are recursive procedures that generate some ill-founded sets in addition to well-founded sets. The interesting question is whether or not the arguments in favour of the more restrictive version – the cumulative hierarchy – were all along arguments for the more general version. 

How to be a minimalist about sets

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical

Complete Totalities

A new approach to sets as totalities is presented based on a notion of “concurrent aggregation,” which instead of avoiding “viscous circles” acknowledges the inherent circularities of some predicates, and provides a way to characterize and investigate these circularities.


A modal set theory is developed which encapsulates this potentialist conception and is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.


  • Tim Button
  • Philosophy
    The Bulletin of Symbolic Logic
  • 2021
Abstract Potentialists think that the concept of set is importantly modal. Using tensed language as a heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets:

Why is the universe of sets not a set?

This paper argues against the two main alternative answers to the guiding question of why there is no set of all sets and outlines a close alternative to the minimal explanation, the conception-based explanation.

The Iterative Conception Of Set A ( Bi-) Modal

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal

Conceptions of Set and the Foundations of Mathematics

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and

Hierarchies ontological and ideological

Godel claimed that Zermelo-Fraenkel set theory is ‘what becomes of the theory of types if certain superfluous restrictions are removed’. The aim of this paper is to develop a clearer understanding of

The Graph Conception of Set

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this

The scope of Feferman’s semi-intutionistic set theories and his second conjecture

The paper is concerned with the scope of semi-intuitionistic set theories that relate to various foundational stances. It also provides a proof for a second conjecture of Feferman’s that relates the



On the Consistency of a Positive Theory

Here it is proved that GPK+∞ + ACWF (ACWF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axio of global choice and theAxiom "On is ramifiable" are mutually interpretable.

On Numbers and Games

ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally

Set theory with free construction principles

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » ( implique l’accord avec les conditions

Set Theory with a Universal Set

Speaking of Objects in: Ontological Relativity and other essays

  • 1968

Church-Oswald models for Set Theory. in: Logic, Meaning and Computation: essays in memory of Alonzo Church, Synthese library 305

  • 2001

Annali della R. Scuola Normale Superiore di Pisa

  • 1889

Speaking of Objects

Logic, Induction and Sets

  • T. Forster
  • Computer Science, Mathematics
    London Mathematical Society student texts
  • 2003
This chapter discusses recursion, set theory, and other topics related to datatypes, as well as some examples of computable functions and examples of partial ordered sets.

On Numbers and Games, 2nd edition

  • 2001