Embedded arithmetic set expressions In set builder notation, we can represent the set: $$\{ 2, 7\}$$ as: $$\{ x | x=2 \vee x=7 \}$$. 

Therefore, the PA arithmetic predicate: $$φ(x) :=  x=2 \vee x=7$$ is capable of representing this set. The boolean expression: $$x=2$$ can also be represented in arithmetic by the function:

$$1-sgn(x-2)²$$

So: $$x=2 \vee x=7$$ is equivalent to: $$(1-sgn(x-2)²) \vee (1-sgn(x-7)²)$$.

Since: $$x \vee y = x+y-xy$$

we can represent:

$$φ(x) := 1-sgn(x-2)² \vee 1-sgn(x-7)²$$

$$φ(x) := 1-sgn(x-2)² + 1-sgn(x-7)²-(1-sgn(x-2)²)(1-sgn(x-7)²)$$

which simplifies to:

$$φ(x) := 1 - sgn((x-7) (x-2))²$$

So, we can have an arithmetic predicate that does not need to use boolean operators and that can represents the set:

$$ \{ a_1, a_2, a_3, ... , a_n \}$$ 

arithmetically as:

$$φ(x) := 1 - sgn((x-a_1)(x-a_2)(x-a_3)...(x-a_n))²$$

Now I am stuck trying to use a similar technique to represent the embedded set expression:

$$\{1, 2, \{ 4, 9\}, 7\}$$

The inner embedded set { 4, 9 } in the expression above may look somewhat like:

$$φ_2(x) := 1 - sgn((x-4)(x-9))²$$

The outer set may look somewhat like:

$$φ_1(x) := 1 - sgn((x-1)(x-2)(x-φ_a(x))(x-7))²$$

But I cannot figure out how to express:

$$φ_a(x)$$ in terms of: $$φ_2(x)$$

Can this actually, uberhaupt, even be done?
Your problem is that your trick of representing set membership using arithmetic only applies to sets consisting of numbers only. Say you have a set of colours { red, green } then 1 - sgn (x - red)^2 doesn’t make any sense. Even worse is a set containing other sets as elements.