Cardinality Theory "CT" is a theory of sets of cardinals and links between them, only sets of cardinals can be assigned cardinalities. The links are unordered edges linking cardinals, they serve to define relations and functions on cardinals. This theory can be seen to be capable of capturing second order arithmetic. The question is whether it can be simply extended to capture ZFC?
Logic: mono-sorted first order logic
$\frak Signature $: "$ =;\in; \overline { \ , \ };|| $"
, the first two are the known binary relations of equality and set membership, the third is a partial binary function denoting linkage, and the last is a partial unary function denoting cardinality.
Define: $*(x) \iff \exists y: x \in y$
Define: $\bigcirc(x) \iff \neg *\!(x)$
$*(x)$ is read as "$x$ is an element", while $\bigcirc(x)$ stands for "$x$ is a set".
Extensionality: $\bigcirc(x) \land \bigcirc(y) \land x \subseteq y \land y \subseteq x \to x=y$
Comprehension: $ \exists x: \bigcirc(x) \land \forall y \, (y \in x \leftrightarrow *(y) \land \varphi)$, if $\varphi$ doesn't use "$x$".
Flatness: $y \in x \to \bigcirc (x)$
Linking: $ \overline {a,b} = \overline{c,d} \leftrightarrow [a=c \land b=d] \lor [a=d \land b=c] $
Links: $* (\overline{a,b})$
Define: ${\large \textbf{-}}\!{\large \textbf{-}} (x) \iff \exists a \exists b : x= \overline{a,b} $
Define: $ \circ (x) \iff *(x) \land {{\large \textbf{-}}\!\!/\!\!{\large \textbf{-}}} (x)$
"$\circ$" is read as "non-linking element"; also can be read as "concrete element". On the other hand "${\large \textbf{-}}\!{\large \textbf{-}}$" is read as "link", and "$x=\overline {a,b}$" is read as "$x$ is the link between $a$ and $b$", it works like an unordered pair, those are seen as a kind of "relational elements", also to be known as "abstract elements". Along the same venue sets of concrete elements are concrete sets, contrasting with the abstract sets of relational elements which serve to implement relations between disjoint sets and can indirectly code some relations between sets in general.
Simplicity: $\exists p: p= \overline{a,b} \iff \circ(a) \land \circ(b) $
Cardinality: $|x| = |y| \iff \exists x_1 \exists x_2 \exists y_1 \exists y_2: x=x_1 \cup x_2 \land y=y_1 \cup y_2 \land x_1 \cap x_2= \varnothing \land y_1 \cap y_2 = \varnothing \land x_1 \cap y_1= \varnothing \land x_2 \cap y_2 = \varnothing \land x_1 \equiv y_1 \land x_2 \equiv y_2$
Where "$\equiv$" denotes existence of a bijective function.
"$x=|y|$" is read as "$x$ is the cardinality of $y$".
Cardinals: $\exists y: x=|y|\iff \circ(x) $
Existence: $\exists x: x=|A| \iff \bigcirc (A) \land \forall y \in A: \circ(y) $
Define: $a \leq b \iff \exists x \exists y \exists z: a=|x| \land b=|y| \land z \subseteq y \land x \cap z=\varnothing \land x \hookrightarrow z $
Where $x \hookrightarrow z$ signify existence of an injection from $x$ to $z$.
Define: $a < b \iff a \leq b \land a \neq b$
Ordering: $ |x| \in A \to \exists l \in A \forall y \, (|y| \in A \to l \leq |y|) $
Define: $\operatorname {limit}(x) \iff \exists n: n < x \land \forall y < x \exists z < x \,( y < z)$
Define: $x=\omega \iff \operatorname {limit}(x) \land \forall y: \operatorname {limit}(y) \to x \leq y$
Parsimony: $ |x| \leq \omega$
A natural is a cardinal strictly smaller than $\omega$. All rules of $\sf PA$ as well as $\sf Z_2$ hold over the naturals and sets of them in this theory. And it does that with a single tier of set membership (flat sets) and the simplest linking pattern forbidding complex linking. So, it has a clear cut simple set and linking structure in which second order arithmetic is implementable!
The following are definitions of addition and multiplication and a global well-ordering. The first two to be applied on the naturals, though can be extended over all cardinals here. The last range over all cardinals.
$a + b=c \iff \exists x \exists y: x \cap y = \varnothing \land a=|x| \land b=|y| \land c=|x \cup y|$
$x \in^G y \iff \forall m \in y \exists a \in G \exists b \notin G \, (m=\overline{a,b} ) \land \exists a \in G: x=\{b \mid \overline{a,b} \in y\} \land \forall a \in G \exists b: \overline {a,b} \in y $
$a \times b= c \iff \exists A: |A|=a \land \exists y: (\forall u\forall v:u \in^A y \land v \in^A y \land u \neq v \to u \cap v = \varnothing \land |u|=b) \land c=| \{l\mid \exists k \in A: \overline{k,l} \in y\}| $
Define: $x \prec y \iff x < y \lor [\circ(x) \land {\large \textbf{-}}\!{\large \textbf{-}}(y)] \lor \\\exists \, a,b,c,d, : x= \overline{a,b} \land y= \overline { c,d} \land\\ \min(\operatorname {proj}(\overline{a,b})) < \min(\operatorname {proj}(\overline{c , d})) \lor \\ [\min(\operatorname {proj}(\overline{a,b})) = \min(\operatorname {proj}( \overline{c , d} )) \land \\ \max(\operatorname {proj}(\overline{a,b})) < \max(\operatorname {proj} (\overline{c,d} ))] $
Global ordering theorem: $ y \in A \to \exists l \in A \forall x \in A :l \preceq x $
So, cardinality theory can interpret $\sf PA$.
Now, my question if we deny Parsimony and add an axiom of Singularity, which states that the maximal cardinal is not singular.
Singularity: $\forall x\, ( \operatorname {singular} (|x|) \to \exists y: |y| > |x|)$
Where "singular" is defined as usual by being the cardinality of a set which is a smaller set union of smaller sets. Formally, for each cardinal $\alpha$ we define $\alpha'$ as the instersectional set of all sets of cardinality $\alpha$ that are closed under $\leq$. Accordingly we have $|\alpha'|=\alpha$.
$\operatorname {singular}(\alpha) \iff\\ \exists x \subset \alpha': |x| < \alpha \land \{z\mid \exists \beta \in x: z \in \beta'\} =\alpha'$
Would that get us to interpret $\sf ZFC$?
