User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-25601061-20180625142705/@comment-35470197-20180627070347

I note that a class \(X\) in ZFC set theory is just a formula with a free variable \(x\) in ZFC set theory, and the sentence \(x in X\) is just a synonym of \(X\) itself.

Moreover, for classes \(X\) and \(Y\), there are many synonyms of formulae in ZFC set theory. \begin{eqnarray*} x \notin X & \stackrel{\textrm{def}}{\Leftrightarrow} & \neg(X) \\ y = X & \stackrel{\textrm{def}}{\Leftrightarrow} & \forall z((z \in y) \leftrightarrow (z \in X)) \\ X \in z & \stackrel{\textrm{def}}{\Leftrightarrow} & \exists y((y = X) \wedge (y \in z)) \\ X \in Y & \stackrel{\textrm{def}}{\Leftrightarrow} & \exists y((y = X) \wedge (y \in Y)) \\ X = Y & \stackrel{\textrm{def}}{\Leftrightarrow} & X \leftrightarrow Y \\ X \subset Y & \stackrel{\textrm{def}}{\Leftrightarrow} & X \to Y \\ \forall x \in X, \Phi(x) & \stackrel{\textrm{def}}{\Leftrightarrow} & \forall x(X \to \Phi(x)) \\ \exists x \in X, \Phi(x) X \subset Y & \stackrel{\textrm{def}}{\Leftrightarrow} & \exists x(X \wedge \Phi(x)) \\ X \cap Y & \stackrel{\textrm{def}}{\Leftrightarrow} & X \wedge Y \\ X \cup Y & \stackrel{\textrm{def}}{\Leftrightarrow} & X \vee Y \\ X \setminus Y & \stackrel{\textrm{def}}{\Leftrightarrow} & X \wedge \neg(Y) \\ z \in X \times Y & \stackrel{\textrm{def}}{\Leftrightarrow} & \exists x \in X, \exists y \in Y, z = (x,y) \end{eqnarray*} I note that the occurence of \(\forall X\) is not allowed in this traditional class convention in ZFC set theory.

For example, a set \(y\), the class \(Y\) given as the formula \(x \in y\) satisfies \(y = Y\). The class \(V\) is defined as the formula \(x \in x\). The class \(\textrm{ON}\) is defined as the formula "\(x\) is an ordinal number".

It is remarkable that such abbreviation is always done in ZFC set theory even if you do not use classes. Remember that the formal language \(\textrm{FOST}\) of first order set theory do not have the function symbol \(\cup\). Therefore for sets \(x,y\), the formula \(z = x \cup y\) is just a synonym of the formula \(\forall w((w \in z) \leftrightarrow ((w \in x) \wedge (w \in y)))\). Then the "set" \(x \cup y\) is unique, i.e. \(\textrm{ZFC} \vdash \forall z(\forall w((z = x \cup y) \to (w = x \cup y)))\) holds. So this abbreviation is never harmful. In order to regard such symbol \(x \cup y\) as actually a set, mathematicians usually implicitly use the notion of "extension of language by definition", which allows us to extend the formal language and the axioms in a completely harmless way. (Such a technique is written mainly in textbooks on logic or set theory, but not in those on other mathematics.)

If many googologists here do not know these conventions, I might be better to write all in my other blog post :D