Axiom talk:Axiom of Specification/Class Theory

Is the second $x \in B$ in
 * $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$

purposeful? It's redundant and isn't in any formulations I've seen --TheoLaLeo (talk) 22:35, 28 November 2021 (UTC)


 * It's in the one in Smullyan and Fitting, who go on to explain:
 * Intuitively, each axiom of $P_2$ says that given any subclasses $A_1, \ldots, A_n$ of $V$ there exists the class $B$ of all elements $x$ of $V$ that satisfies the condition $\map \varphi {A_1, \ldots, A_n, x}$. We remark that $P_1$ is to hold even when $n = 0$, i.e. even when there are no class variables in the formula.
 * --prime mover (talk) 22:50, 28 November 2021 (UTC)


 * In the 1996 edition he has:
 * $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \land \phi {A_1, A_2, \ldots, A_n, x} }$
 * not:
 * $\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$
 * Unless the second $x \in B$ is supposed to be $x \in V$ I think it's redundant


 * I think the part about there being no hidden bound proper class variables should be explicit.
 * $\phi {A_1, A_2, \ldots, A_n, x}$
 * is not required to have no variable inputs other than $\paren {A_1, A_2, \ldots, A_n, x}$. For instance, if you wanted $\phi x$ to be "x is the limit of some sequence in set $S$" the $\delta, \epsilon$ in the limit definition would be hidden variables that are bound inside the $\phi$, not outside as one of $\paren {A_1, A_2, \ldots, A_n}$. I think highlighting that there can be no hidden bound proper class variables is fairly important bc it can be kind of subtle to catch when working with high level definitions that mask the primitive symbols they actually use, and it is a fundamental part of the axiom's statement. The Morse-Kelley could be an also see link to another page or something --TheoLaLeo (talk) 23:17, 28 November 2021 (UTC)


 * Revisiting this ... you are completely correct, I misread it. Now I notice that extra $x \in B$. Sorry, I need to concentrate on one thing at a time rather than try to do more than one thing at once.


 * Fixed it. --prime mover (talk) 23:37, 28 November 2021 (UTC)