Sets and supersets

Abstract

It is a commonplace of set theory to say that there is no set of all wellorderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe (Halmos, in: Naive set theory, 1960). In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.Keywords Philosophy of mathematics · Philosophy of set theory · Formal theories of truth · Absolute generality · Indefinite extensibility You're gonna need a bigger boat.(Jaws)The idea that there ought to be a set containing all of the sets is the focus of this paper. It is not a new idea. Moreover, I suspect that my way of addressing this issue will have occurred to many readers, although usually they they will have set it aside. I do not want to suggest that the techniques explored in this paper are of themselves particularly original; we shall see that related approaches have been explored from a technical angle for many years. Rather, what I am aiming to do is give this group of ideas a more philosophical twist by drawing out their underlying motivations and B Toby Meadows

Plurals and Mereology

“…super-set and super-membership) to give an analysis of talk of sets and members in the plural. This strategy has been recently developed and defended in Meadows [49] (see also Williamson [82]). This formula is not only consistent but also true if one adopts the appropriate axioms for super-sets.…”