Structure Theory:

FOL+ID+GEAM+

[1] (f|a->b)^(g|c->d) ->[f=g <->(a=c^b=d)] 

[2] a,b are atoms -> Exist f. atom(f)^(f|a->b) 

[3] (f|x->y) -> x,y are atoms 

[4] V* exists.

[5] X < V -> P(X) < V 

[6] Exist I: I is infinite & I < V. 

[7] X P V* ^ X <* V* -> X < V
/


To explain it:

ID is "Identity Theory"

GEAM is General Extensional Atomic Mereology (without bottom).

P stands for 'is a part of' relation.

f|x->y is phrased as: f is an arrow from x to y; also x,y are said to be the  
'arguments' of arrow f, also x is said to be the "origin" of arrow f while y  
is the target of arrow f.

A structure is any object in this theory!

A structure shall be labeled "Small" if and only if it is strictly subnumerous  
to the totality V of ALL atoms, we'll use the symbol "<" to denote strict  
subnumerousity of 'atoms', so x < V would be taken to mean "x is strictly  
subnumerous to V", i.e. x has less atoms that do V have.

A structure A is said to be "in touch" with structure B iff A and B are non  
overlapping and: there is an atom of A that is an argument of an arrow of B;  
or there is an atom of B that is an argument of an arrow of A. 

Now a structure x is said to be Continuous if and only if every non  
overlapping parts y,z of x are either in touch or there exist a part k of x that  
is in touch with both of them. 

Of course a Discrete structure is a non-continuous structure. 

X is a moiety of Y iff 
X P Y ^ X is continuous ^ for all Z. Z P Y ^ Z is continuous ^ Z O X -> Z P X. 

"O" signify "Overlap" as standardly defined in Mereology. 

Now the basic idea behind structure theory is to try to define as much  
structure as we can. Of course two structures are said to be "similar" iff they  
are "isomorphic". 

A "Power structure" of any structure X is a structure P(X) in which every  
moiety is similar to some substructure of X; and such that for every continuous
substructure Y of X there is exactly one(up to identity) moiety of P(X) 
that is similar to Y.

Define: a V* structure is a tructure in which every moiety is isomorphic to 
some small structure AND in which for every small continuous structure there
is exactly ONE(up to identity) moiety that is isomorphic to it. 

We can use V* to simplify defining powers, so P(X) become the totality of all  
moieties of a V* that are isomorphic to parts of X.

The relation X <* Y is existence of one-one relation from moieties of X to  
moieties of Y and the absence of one-one relation from moieties of Y to 
moieties of X.

Most probably this theory can interpret both Set Theory (ZFC) and Category  
Theory (Topos) much easier than the other two interpreting it (if they can), so it  
might provide easier foundation for wider variety of structures without the  
need to squeeze interpreting them after a particular form of structure.

Well founded sets are interpreted as directed mono-rooted trees with all arrows 
directed away from the root node, set members are interpreted as being  
immediate mono-rooted subtrees of a tree, i.e. those that have their root  
nodes targeted by arrows from the root node of the tree provided that no  
subtrees originating from their root nodes can have them as proper parts of.

Categories are much easily definable in almost straightforward manner.

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I'd advocate Structure Theory as a more elegant way of founding mathematics!
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Zuhair

To directly interpret sets in structure theory here I'll present a simple structure theory that extends General Extensional Atomic Mereology by adding the primitive ternary relation "is an edge from to" with rather very simple axioms and structural rules that seems to be very natural. Basic axioms about edges and nodes states that x is an edge from a to b entails that x,a,b are atoms, a and b will be called "nodes" while x is an "edge"; an axiom asserting the existence of at least one edge between any two atoms; also an axiom stating that no edge is a node; of course each edge has unique nodes that it strands from to. Now we define a nice tree as a tree having no infinite path and in which no node is a root node of two isomorphic subtrees of it. Each nice tree would correspond to a well founded extensional set. A nice forest of trees would be a totality of nice trees where no two FREE (not part of another tree in the forest) trees are isomorphic to each other. 

Now the rules of forming nice trees from nice trees would be: 

Rule 1: There is a nice forest F! in which every tree have an isomorphic FREE tree in F!. 

Rule 2: For every nice tree t there is a nice power tree. 

Rule 3: Every nice tree t and a nice forest of trees F if there is one-one relation between nodes of a part of t and the root nodes of trees in F, then a nice tree can be formed from replacing the end nodes of t by the those free trees in F. 

Rule 4: There is an infinite nice tree 

/ 

All those rules are actually almost naive, Rule 1 signals (BREAKING) operation which is a pure structure tool, while Rules 2 and 3 are COMBINING rules, the power rule is just saying that all broken parts from a nice tree that are trees can be combined to form a nice tree, which is plausible since all those parts already belong to a single nice tree so it is plausible to think that we can recombine all of those subtrees into a tree which would be nice, just merge the root nodes of all those broken parts (free trees in F! that are isomorphic to subtrees of the nice tree) and you get a nice power tree, so naive construction theme. On the other hand the replacement of end nodes of a nice tree by nice trees (just merge the root nodes of those nice trees with the end nodes of that tree); those two combining rules are just simple root to root merging and end by root merging which result in a nice tree structure (while end-end merging result in cyclic structures thus not trees). So there is a clear structural naive combining theme behind those rules. 

As regards infinity this is just to meet the urge for going beyond the finite world, and it doesn't seem to inflect any of the above rules. 

I consider that as constituting some motivation for axioms of ZF. So structure theory especially of trees can provide a nice interpretation for sets. 

Of note there is no need to insist on the trees being nice, one can generalize the above axioms over all kinds of trees. I would also hasten to say that one can generalize the above axioms over all kinds of CONNECTED DIRECTIONALLY Uniquely Edged Graphs, "CDUEG", And what I mean by CDUEGs are graphs in which between each two distinct nodes of there is no more than one edge between them per direction, and of course each CDUEG is a continuous structure, i.e. not a forest of separate graphs. I think that extension would interpret CATEGORY THEORY as well. So this mean that such extended graph theory would interpret both SET THEORY as well as CATEGORY THEORY in one go.

Clearly Category theory would be easily grounded in structure theory (a theory about node-edge graphs), and here I've presented a grounding of Set theory in structure theory. Thus it appears to me that "Structure Theory" is the ultimate foundation of mathematics. From the womb of structure theory many foundational theories like: Set theory, Category theory, and perhaps other particular structure oriented theories, may rise to day light. 

Regards 

Zuhair Al-Johar
April 1 2015