Circular logic in set definition - Tautology? How useful is a definition of this form: $C=\{\beta\,:\,Q(\beta,\gamma)=0,\,\gamma\notin C\}$? (Q is some function of $\beta$ and $\gamma$.) Would this not be an instance of circular logic or does it indeed pin down the $\beta$'s uniquely? In a discussion, I was made aware of the fact that it's perfectly legitimate to use the orthogonal complement in a definition of a set but to me it seems like a tautology. Any comments on this would be much appreciated! Thank you.
 A: If $Q$ is a function such that $Q(\beta,\gamma) = 0 \iff \beta = \gamma$, then your definition comes down to
$$
C = \{\beta \mid \beta \notin C\},
$$
which is not a valid definition.
A: This is not allowed by the axioms of ZFC. And you cannot allow it in any classical set theory, because it gives a contradiction.
$
\def\nn{\mathbb{N}}
\def\wi{\subseteq}
$

Let $Q(x,y) \overset{def}\equiv \cases{ 0 & if $x = y$ \\ 1 & otherwise }$.
Let $C = \{ x : Q(x,y)=0 \land y \notin C \} = \{ x : x = y \land y \notin C \}$
Then $C \in C \equiv \exists y\ ( C = y \land y \notin C )$.
If $C \in C$:
  Let $y$ be such that $C = y \land y \notin C$.
  Then $C \notin C$.
If $C \notin C$:
  Then $\exists y\ ( C = y \land y \notin C )$.
  Thus $C \in C$.
Contradiction.

In fact, even if you require $C \wi \nn$ and $Q$ to be a binary function on $\nn$ rather than a function-like formula, it can still be used to get a contradiction.

Let $Q(m,n) \overset{def}\equiv \cases{ 0 & if $m = n$ \\ 1 & otherwise }$, for each $m,n \in \nn$.
Let $C = \{ m : Q(m,n)=0 \land n \notin C \} = \{ m : m=n \land n \notin C \} = \{ n : n \notin C \}$.
Then $0 \in C \equiv 0 \notin C$.

Note that this second version seems shorter but it really requires the same reasoning as the first in order to prove the chain of equalities.
