Has anybody investigated or constructed a set theory, call it $T$, which develops in a manner that is the reverse of how the majority of the standard set theories develop? Instead of starting with the null set, generating the finite sets and expanding into larger and larger infinite sets, either endlessly or until a universal set is reached, the theory $T$ would proceed in the opposite direction. $T$ starts with a universal set and each set (in the universe) of $T$ generates a collection of proper subsets. This descending process either continues endlessly or culminates in finite sets, singletons-and finally a null set. The primitive notion of $T$ might be "$x$ is a proper subset of $y$" instead of "$x$ is an element of $y$" (which is the primitive notion of the standard set theories). Could there already exist such a theory as $T$, that might be of any interest?
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$\begingroup$ It seems to me like you want to ask if the cumulative hierarchy can be inverted in the sense that you explained. $\endgroup$– Git GudCommented Aug 14, 2014 at 19:32
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$\begingroup$ If your set theory allows construction of subsets with a given condition (which is part of ZF), then any set allows you to construct the empty set: $\emptyset=\{x\in X;\text{false}\}$. $\endgroup$– Joonas IlmavirtaCommented Aug 14, 2014 at 19:44
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$\begingroup$ I am asking if such a set theory as T already exists or has been constructed by somebody. I do not Know if it is even possible to construct-even partially-a theory of this kind $\endgroup$– Garabed GulbenkianCommented Aug 14, 2014 at 21:13
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$\begingroup$ Naively, if you wish to start at the "top" and work your way down, you would need to define the "top". The cumulative hierarchy, for example, has no top. Any set theory which may arise from such a construction would need to be closed under the power set operation, assuming the power set is definable in your theory. I believe that closure under the power set operation is a large cardinal property, so you will need to start with something very big. $\endgroup$– gammaCommented Aug 15, 2014 at 0:26
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$\begingroup$ The "top" would be something like the universal set in Quine's set theory NF. I believe this has recently been proved consistent with ZF. But the difficult part of this problem is how to come down from this "top". For example, certain sets-but not all-are the power sets of other sets. How to pick out the ones that are? $\endgroup$– Garabed GulbenkianCommented Aug 17, 2014 at 18:23
3 Answers
One possible way to build sets iteratively from "the universe" instead of the empty set (and the process if fairly symmetrical to the usual one) is to replace axioms with duals. Instead of the axiom of extensionality
$$X = Y \equiv (t \in X \equiv t \in Y)$$
use the axiom of "intensionality", according to which, sets belonging to the same sets are equal:
$$X \sim Y \equiv (X \in t \equiv Y\in t),$$
instead of comprehension-based set builder $\{x\ |\ P(x)\}$
$$t \in \{u \mid P(u)\} \equiv P(t)$$
use, hmm.. "incomprehension-based" set builder $[x\ |\ P(x)]$
$$[u \mid P(u)]\in t \equiv P(t),$$
etc. Shameless plug.
Mereology could be described as set theory with the subset relation as the primitive notion. Check out http://plato.stanford.edu/entries/mereology/. The works by Stanisław Leśniewski in the references might contain what you are looking for.
This paper - http://www.pgrim.org/philosophersannual/pa28articles/forsterset.pdf - by Thomas Forster describes iterative concepts of sethood which are broader than the classical hierarchy, and spends most of its time discussing an approach which builds both upwards and downwards simultaneously.
