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Working in mono-sorted first order logic, add primitives of equality and its axioms, set membership $\in$, a partial ternary relation $\to$ denoting is the direction from to, and at last a total unary function $\cal S$ dentoing the "structure" of.

Axioms for sets:

Extensionality: $\forall x \forall y \ (\forall z (z \in x \iff z \in y) \implies x=y)$

Membership: $x \in y \implies \forall z \in x (z =x)$

Atomicity: $\forall x \exists y: y \in x$

Comprehension: $[\exists y (y=\{y\} \land \phi)] \implies \exists x \forall y \ (y \in x \iff y=\{y\} \land \phi)$; for every formula $\phi$ in which $x$ is not free.

Axioms about direction:

Naturality: $\to(x,a,b) \implies x,a,b \text{ are singletons }$

Uniqueness: $\to (x,a,b) \ \land \to(y,c,d) \implies [x=y \iff (a=c \land b=d)]$

Incompatibility axioms:

Define: $node(a) \iff \exists x \exists b :\, \to (x,a,b) \ \lor \to (x,b,a)$

Define: $arrow(x) \iff \exists a \exists b :\, \to(x,a,b)$

Dichotomy: $node(a) \implies \neg arrow(a)$

Duality: $ a=\{a\} \implies node(a) \lor arrow(a)$

Axioms about size:

Definitions:

A graph is a set closed under node-hood. That is, all nodes connected by arrow elements of it are elements of it.

A scatter is a set of nodes only.

The scatter of a graph is the set of all of its nodes.

"Nodes" is the set of all nodes.

A graph is to be called small if and only if its scatter is strictly subnumerous to Nodes.

Where "strictly subnumerous" is defined in the cusmtomary manner after existence of injections in only one direction.

Two graphs are said to be separate if there is no arrow connecting a node from one of them to a node in the other.

A graph is continuous if and only if it's not the set union of two separate graphs.

A moiety of a graph is a maximal continuous subgraph of that graph. That is, no continuous subgraph of that graph exists that has it as a proper subgraph of.

Axioms:

Infinity: There is a small infinite graph.

Power: for every small graph there exists a small power graph.

The power graph of $G$ is one whose moieties are isomorphic to subgraphs of $G$, having no two distinct moieties being isomorphic to each other, and where every subgraph of $G$ is isomorphic to some moiety of it.

Inaccessibility: The small fusion of small separate graphs is small.

In other words if a graph have less many moieties than the nodes in Nodes, and each moiety is small, then that graph is small too.

Axioms about structure:

Abstraction: $\forall \ graphs \ x,y: x \approx y \implies \cal S(x) = \cal S(y) $

Canonicity: $\forall \ graph \ x: \cal S(x) \approx x$

Define: $x \approx y \iff \exists f:f \text{ is isomorphism from } x \text{ to } y$

Where:

$f \text{ is isomorphism from } x \text{ to } y \iff \\ f \text{ is a bijection from } scatter (x) \text { to } scatter (y) \land \\\forall a,b \in x [(\exists k \in x :\to(k,a,b)) \iff \exists l \in y :\to (l,f(a),f(b))] $

Separateness: $G, H \text{ are small moieties } \implies \mathcal S(G) \ disjoint \ \mathcal S(H) $

Axioms about choice:

Choice: for every graph $G$, there exists a scatter graph that contains exactly one node from each moiety of $G$.

The above structure theory does interpret ZFC and actually MK. The category of all sets of ZFC is definable as the set of all structures of small extensional mono-rooted trees with finitely long branches, plus all arrows between those structures including the identity arrows over their nodes. By extensional tree, it's meant that no node of it can have two distinct isomorphic maximal subtrees stemming from it. The set membership relation of ZFC can be defined over those structures as structures of maximal subtrees whose root nodes are those connected to the root node of the main tree directly through arrows. So it does provide an explication about sets and their membership in the standard sense of ZFC. I believe also that the category of all small categories can as well be defined here in almost straightforward manner. So, this theory can serve as a natural foundation of both Set and Category theory.

Can we regard such a theory as a Candidate for a foundational theory of mathematics?


Technical development about size expressions:

The $\phi$-cardinality of $x$ is the number of $\phi$ parts of $x$, this occurs when any two parts of $x$ satisfying $\phi$ are separate [disjoint]. Now this is determined by existence of a one-to-one relation $R$ from $\phi$ parts of $x$ to nodes of a scatter $k$, that is, all arrows in $R$ that are sent from nodes of $x$ to an element $j$ of $k$ all of them come from nodes in a single $\phi$ part of $x$, and of course for every node of $k$ there is an arrow in $R$ that comes from a node of $x$ to it, and to assure the one-to-one genre we must have distinct nodes in $k$ having nodes beloning to distinct $\phi$ parts in $x$ with arrows coming from those to them. Now the $\phi$-cardinality of $x$, symbolized by $|x|^\phi$ would be equal in this case to $|k|$.

More generally, if we have a binary relation $Q$ and an object $x$, then $|x|^Q$ is the cardinality of all objects that bear the relation $Q$ to $x$ provided that all those objects are separate! This would have the same above definition, we only replace the relation $part$ by $Q$.

Now we'll define cardinality generally as the structure of a scatter graph, and so it reflects how many nodes are there in scatter graphs. So formally

Define:$ |x|= y \iff scatter(x) \land y=\mathcal S(x)$

So the node-cardinality of any graph is defined as:

$|x|^{node} = y \iff \exists k \exists R: scatter(k) \land R: scatter(x) \to k \land R \text{ is a bijection } \land |k|=y$

This will boil down to:

$|x|^{node} = |scatter(x)|$

Now

$|x|^{moiety} = y \iff \exists k \, \exists R: scatter(k) \land R \text{ is one-to-one from moieties of } x \text{ to nodes of } k \land y=|k|$

So, the last two size axioms cab be formally written as:

Power:$|x|^{nodes} < |Nodes| \implies |x|^{substructure} < |Nodes|$

Inaccessibility: $|x|^{moiety} < |Nodes| \land \forall y : y \text{ moiety of } x \to |y|^{nodes} < |Nodes| \\\implies |x|^{nodes} < |Nodes|$

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  • $\begingroup$ Any theory can be a candidate for a foundational theory — why would this be a good candidate? $\endgroup$
    – user44143
    Jul 16, 2021 at 23:03
  • $\begingroup$ @MattF. This theory can interpret both set theory and Category theory, and in a very natual manner, its primitives are very clear, and its axioms appears to naturally capture them. $\endgroup$ Jul 17, 2021 at 8:55
  • $\begingroup$ I think there is dispute what qualifies a theory as a foundational theory of mathematics. Perhaps your question should first be more along the lines, What qualifies a theory as foundational (which may well be downvoted by some because it is not a mathematical question)? And then, does this particular theory have those characteristics? $\endgroup$
    – abo
    Jul 18, 2021 at 15:12

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