"Not in set" notation within the set definition

Question

I am trying to define a set such that for two elements $a$ and $b$, such that $a$ can only be in the set if $b$ is not. For example, lets say $a$ is a natural number and $b = 5 - a$. I defined it as $S = \lbrace { a | \exists b: (a + b = 5) \wedge a \in \mathbb{N} \wedge b \notin S}\rbrace$. My question is, can I use S on the right hand side of the definition of S itself?
(The example is very trivial. It is just for clarity.)

Answer 1

Is $3$ in the set? Only if $2$ is not.

Is $2$ in the set? Only if $3$ is not.

How do you know whether either $2$ or $3$ is in $S$? You can't, not without knowing whether either $2$ or $3$ is in $S$,

Hence for example either $S = \{0, 1, 2\}$ or $S = \{3, 4, 5\}$ could be $S$. And all sorts of other possibilities, e.g. $S = \{0, 2, 4\}$.

So $S$ is not uniquely defined.

Hence your definition is invalid.

In fact you can even make a case that $S = \{0, 2, 4, 6, 7, 8, 9, 10, \ldots\}$ because you can say, oh yes, but there exists $b$ as a negative integer.