This is how I have constructed a set of predicates for an object $x$. However, I am not sure how correct this construction is, or if it is, at all possible, to construct a set of predicates. So, in essence, I am trying to build a set $A$ whose members are all the predicates of an object $x$.
$$A=\{\varphi|\varphi:x\to\text{{true}}\}$$
Now, I want to to go a step further and say:
$$B=\{\rho|\rho:A\to\text{{true}}\}$$
Here, I am trying to define the Set $B$ whose elements are all the predicates of the set $A$.
How good are the formalisms, anything missing?
I think people are still having a hard time understanding what I am trying to ask. I am saying is it possible to create a set $A$ such that whose elements are all the predicates of a particular object $x$. $P_{1}...P_{n}\in A$, where $P_{1}...P_{n}:x\to\text{{True}}$
You can do something like this if you use type theory. It comes in many versions, and isn't as popular as set theory. If $x$ is a first-order object, the predicates it satisfies are second-order, and the collection of them you call $A$ is third-order. By the same logic, $B$ is a fifth-order object; so you'll need a type theory with at least five types (not all have infinitely many).
There are several alternative approaches. You can take a theory of order $<5$, then add enough metalanguage layers to get the complexity you need. Or you could slightly modify your ambitions, and get by in a first-order set theory. How? We could fix a set $x$ and define $A$ as the set of rank-minimal sets of which $x$ is an element, and $B$ as the set of rank-minimal sets of which $A$ is an element. (You can then prove $\{ x\}\in A,\,\{ A\}\in B$.) That might not be exactly what you intended, but at least it's somewhat consistent with what most foundational mathematics does. For example, a common definition of the cardinal of a set $x$ is the set of rank-minimal sets that can form a bijection with $x$. The rank constraint is used to ensure the result is a set.
If you'd prefer a type=theoretic approach, be warned it has its own meaning for "rank".