# Is a top-down set theory possible?

There are two classical paradoxes associated with set theory; and that is the existence of the Universal set and the Russell set.

The usual set theory takes the notion of element as basic; these are the atoms by which sets are built.

In the debate between Aristotle and the atomists; Aristotle took the tack the universal divisibility isn't possible ie atoms do not exist.

Could one take a similar tack with set theory - that this is possible as a coherent strategy is shown by the existence of Category Theory ie there are categories without global elements.

That is start from the supposition that there is a universal set; and see what can be drawn from this?

Ie a set-theory from a top-down perspective rather than a bottom-up one.

(If such a theory could be constructed; it would be ironical, and apposite (or perhaps, opposite) if elements were shown to be inconsistent).

It seems to me that propositional logic itself, as usually used, is a top-down set theory. Instead of starting with elements, it begins with restrictions, and it may never converge upon an element.

Many instances of real mathematics really work from propositional axioms, and are only incidentally about elements. The 'elements' are totally ambiguous anyway, since the sets only exist up to isomorphism. (Geometry is not about points. And Group theory is about groups, not their elements.) Category Theory captures this really well, but from a more Universal Algebra perspective, in the vast majority of instances it is an artifact of the nature of axioms, and so of propositional logic.

To the degree one can use (consciously or otherwise) Nonstandard Analysis of a Universal Algebra, basically without missing a beat, one is really handling only potential objects, some of which just happen to be real.

This is one reason why the independence of the Axiom of Choice, which seems merely amusing to many, mattered so much to some mathematicians. You want to handle rules and pretend you are talking about elements, but that is only true if you can just whip out an element whenever you want, as long as you know the rules have not ruled one out. When you need to be less cavalier about how propositions and elements fit together, the nature of the mathematical process changes.

• Interestingly Quines New Foundations disproves Choice. – Mozibur Ullah May 2 '15 at 15:33
• Right, he basically guts Choice by including a selective hobbling of Zorn's Lemma as an axiom: Another way of defining weak stratification is "Any partially ordered set is only well ordered if determining the referents of its self-referential definitions in different ways does give different interpretations of the partial order." Knowing he did it on purpose makes it less interesting. – jobermark May 2 '15 at 15:53

I don't know if this quite answers your question, but there is a field of mathematics called Matroid theory that seems related: in Matroid theory, the fundamental idea is the notion of an independent sets. (One can think of Matroid theory as a generalization of the idea of dependent set of vectors from linear algebra and cycles in graph theory.) Elements come in to play because the relation between them determines which sets are independent, but the important questions are about the sets of elements themselves.

Of course, a matroid as defined as the set of all sets that have certain properties, so even in this case the sets themselves are just elements of another set.

• "Elements come in to play because the relation between them determines which sets are independent" — sounds accurate to me, but that more or less makes it a bottom-up theory. – Niel de Beaudrap May 2 '15 at 5:04