0
$\begingroup$

Problem: "Show the intersection of a class and a set is a set;" and it comes from material covering the axiom schema of comprehension.

I'm using the axiom schema of comprehension as it's listed on Wikipedia and my class notes; that is,

$$\forall w_{1},...,w_{n} \forall A \exists B \forall x (x \in B \iff [x \in A \land \varphi(x,w_{1},...,w_{n})]),$$ or in words, "Given any set $A$, there is a set $B \subseteq A$ such that, given any set $x$, $x \in B$ if and only if $x \in A$ and $\varphi$ holds for $x$."

My attempt at a solution:


Let $S$ be a set. Let $C= \{ x| \varphi(x,y_{1},...,y_{n}) \}$ be the class containing all sets $x$ such that the formula $\varphi(x,y_{1},...,y_{n})$ holds. By the axiom schema of comprehension, for our formula $\varphi$ and our arbitrary set $S$, there exists a set $T \subseteq S$ such that for all sets $z$, $$z \in T \; \text{ iff } \; z \in S \text{ and } \varphi(z,y_{1},...,y_{n}).$$ Define $T= \{s \in S | \varphi(s,y_{1},...,y_{n}) \}$. If $T= \emptyset$, then $T$ is a set. If $T$ is nonempty, then there exists some $t \in T$. Since $t \in T$, we have $t \in S$. Note also that we have $t \in C$ by definition of $C$ (that is, $t$ is some set fulfilling our formula $\varphi$). Hence $t \in T$ if and only if $t \in S$ and $t \in C$. Therefore the set $T \subseteq S$ is the intersection of the arbitrary set $S$ and arbitrary class $C$.


My problem is that while understanding the axiom schema of replacement is one thing... using it absolutely correctly is something else (unfortunately, most advanced math books don't come overly equipped with examples on these sorts of things.)

A main concern is that I properly employed the formula $\varphi$. Was I correct in assigning it to some arbitrary class $C$, and then saying 'for any set $S$, there must also be a subset $T$ for which $\varphi$ holds true for every member of $T$'?

To any set theory minded folks, what improvements would you make? How you would grade such a solution? Thanks!

$\endgroup$
0
$\begingroup$

Your reasoning looks good to me, but a couple of questions: 1. What kind of system of Set Theory are you working with in your course? ZF, Von Neumann-Bernays-Gödel? 2. Have you formally defined the intersection of a class and a set?

I see no problem in defining an arbitrary class by an arbitrary formula, and so to use this same formula as to define a subset of an arbitrary set. Good job!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.