What determines internalization of graph-structures into the set world?

Can we add to $$\sf MK$$ a cardinality function that is indexed by sets?

That is, add a new primitive total unary function symbol $$C$$, then axiomatize: $$C(X) = C(Y) \leftrightarrow X \cong Y \\ \forall X: C(X) \in V$$, where $$V$$ is the class of all sets, and $$\cong$$ is equinumerousity.

$$\sf MK$$ here is the mono-sorted version. The idea is for $$C$$ to be useable in Comprehension and Replacement.

If we define graphs as classes of nodes and edges such that any edge in the graph must have the nodes it connects in the graph also.

A node can be captured as a triple singleton, $$\iota^3x$$, while the edges are the Wiener pairs $$\langle a,b \rangle$$ Formally:

$$\operatorname {Graph}(G) \iff \\\forall x \in G \, ( \exists y: x=\iota^3y \lor \exists a \exists b: x= \langle a,b \rangle) \\ \land \forall a \forall b: \langle a,b \rangle \in G \to \iota^3a \in G \land \iota^3b \in G$$

A structure is a function that index isomorphism between graphs.

A cardinal can be viewed as a structure of edgeless graphs, i.e. scatter graphs.

Here in the above context we say cardinality is internalized into the set-world, because its indexed by sets. So, we can speak about cardinality of proper classes from within the set world.

Now, the question is: What properties should graphs have in order for them to have their structure internalized into the set-world? Should they be non-continuous graphs for example?

• What's wrong with having $C(x)=\vert x\vert^+$ in the usual sense for $x\in V$, and $C(y)=\emptyset$ for $y$ a proper class? Commented Jun 3 at 23:55

The answer is yes for equinumerosity (provided...), but no for graphs.

Equinumerosity. KM is commonly taken to include the axiom of choice and furthermore the axiom of global choice, and this makes things very easy in the equinumerosity case, since we can just let $$C(X)$$ be the smallest ordinal that is equinumerous with $$X$$, if $$X$$ is a set, and otherwise some non-cardinal default value for the proper classes, since these are all equinumerous under global choice.

If you drop the axiom of choice, then it is still possible to get a solution to the cardinal-assignment problem for sets only, that is, when $$X$$ is a set, by defining $$C(X)$$ to be the set of minimal-rank sets that are equinumerous with $$X$$. This is an equinumerosity invariant as desired. (And in fact, this application is the original use of Scott's trick, solved by Scott as an undergraduate at Berkeley in response to a question posed to the class by Tarski.)

Without global choice, I believe it is not in general possible to find a solution to the cardinal assignment problem, since there could simply be more equinumerosity classes amongst the proper classes than there are sets. I'm less sure about this now, but I believe that a class analogue of a suitable symmetric model construction might produce a model with more than $$V_\kappa$$ many class equinumerosity classes. I am unsure if this works with KM as opposed to mere GB, and I would welcome someone posting about it. (See this related question, still unanswered.)

Fregean abstraction. Meanwhile, there is a very general problem here regarding what is known as Fregean abstraction, where one wants to assign abstractions with respect to various general equivalence relations. The problem has many variations, depending on whether the field of the equivalence relation is just sets or also classes, and whether the equivalence relation is first-order definable or second-order definable and so forth.

I mounted a very general analysis in my paper:

The paper has several theorems providing solutions to the Fregean abstaction problem in many natural variations.

Graphs. In particular, there is even in ZF a definable operation on graphs (of set size) that is a graph-isomorphism invariant. Just map every graph to the set of minimal-rank graphs that are isomorphic to it.

One can extend this to class-sized graphs, in the context of GB where every class is first-order definable, using the methods of my paper.

In KM, however, there can be no map $$X\mapsto C(X)$$ that assigns to every class graph $$X$$ a set object $$C(X)$$ that is an graph-isomorphism invariant. The reason is that every class $$X\subseteq V$$ is determined by a certain canonical directed graph that encodes the hereditary $$\in$$ structure on $$X$$, and this digraph can be canonically coded with a graph. So the assignment would provide an injective mapping of the classes into the sets. But KM refutes the existence of such an injection, using the usual Cantorian diagonalization.

• I'm interested in internalization of structure of class-sized graphs! So, if these are first order definable, then we can internalize their structure! Is that correct? Also in MK we can do the same but for definable graphs from set parameters by bounded in $V$ formulas. Can the abstraction function doing that be useable in comprehension and Replacement of MK? Commented Jun 4 at 21:10
• One issue is that, like equinumerosity, the isomorphism relation of proper class graphs is not a first-order property, but second order, and this places you into the more difficult cases considered in my paper. But in models where all classes are first-order definable, you will achieve the results of theorem 9, showing that the invariants are second-order definable. But lastly, models of KM never have all classes first-order definable, and so this result is really about GBc, or about augmenting ZFC models with their definable classes. Commented Jun 5 at 13:06
• And incidently, dealing with the issue of whether all classes are (first-order) definable and the subtleties of these models and their variations is a case where the two-sorted approach to GB and KM seems to make things much cleaner, as opposed to the one-sorted approach that you often use. Commented Jun 5 at 13:59
• In reality I do like the duality you assert in your article and here. But, I have more radical views about it. I think the true foundational ontology is of Atomic General Extensional Mereology without Bottom. The set world to me is the Mereological totality $A$ of all atoms. If a formula is purely written in the set-world (all quantifiers bounded to $A$, and parameters too), then this is like first order definable notion. I call it set-definable. Otherwise it is class-definable (what you call second order). Classes ARE objects but they may not be atoms, that's why they may not be members. Commented Jun 5 at 15:15
• OK, but meanwhile, my objection would be that that single-sort foundational framework obfuscates many central issues, and therefore fails at one of the purposes of being a foundation. Commented Jun 5 at 16:30