the graph conception of set - Google Sites

T HE G RAPH C ONCEPTION OF S ET L UCA I NCURVATI∗ Notice. This is the preprint of an article that appeared in The Journal of Philosophical Logic. The final publication is available at Springer via http://dx. doi.org/10.1007/s10992-012-9259-x A BSTRACT 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 neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA is justified on the conception, which provides, contra Rieger (2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation.

As to classes in the sense of pluralities or totalities, it would seem that they are likewise not created but merely described by their definitions and that therefore the vicious circle principle in the first form does not apply. I even think there exist interpretations of the term “class” (namely as a certain kind of structures), where it does not apply in the second form either. Kurt G¨odel (1944: 131) Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) has an axiom of Foundation. The axiom states that every non-empty set A has an element disjoint ∗ Faculty

of Philosophy & Magdalene College, University of Cambridge, [email protected]

1

from A, i.e. that the membership relation on any family of sets is well-founded. In the presence of the other axioms of ZFC, the axiom is equivalent to the assertion that there are no infinite descending chains of membership, that is chains of the form X0 3 X1 3 X2 3 . . . Hence, the axiom rules out the existence of, for instance, any set A such that A ∈ A and of closed membership chains such as X0 ∈ X1 ∈ X2 ∈ X0 . But not only does the axiom rule out the existence of certain sets; it does so in such a way as to give rise to a picture of the set-theoretic universe as a hierarchy divided into levels. For let us define the levels Vα of the cumulative hierarchy of sets as follows (where α is any ordinal): Vα =

[

P(Vβ ).

β