Seemingly circular definition of a collection

Question

In the book I'm reading on set theory right now, I'm given this definition:

"If $\Phi(x)$ is a formula then the term $\iota !y(\forall x)(x\in y\iff \Phi(x))$ is abbreviated $\{x:\Phi(x)\}$ and read 'the collection of all $x$ such that $\Phi(x)$'."

Later, it is stated that:

"We have defined 'collection of...' but not 'collection'...The naive idea is that collections are precisely objects of the form $\{x:\Phi(x)\}$ for some formula $\Phi(x)$. This does not work under the formal restrictions we imposed in the Introduction. [Only first order logic] Instead:

We say that $a$ is a collection if $a=\{x:x\in a\}$".

Now maybe I'm misreading this, but it appears to me that the definition of a collection as shown here is circular. It seems that we have "$a$ is a collection if it is the collection of all $x$ that belong to $a$"...which is not at all illuminating.

Am I misreading this?

Answer 1

My references are to the revised edition :

• Michael Potter, Set Theory and its Philosophy (2004).

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Potter has "revived" the (not very usual today) treatment of set theory with individuals or atoms (also called : ur-elements).

Loosly speaking, atoms are objects of the domain of the interpretation that can be members of collection but that has no memebers, i.e. if$x$ is an atom, then for no $y$ we have $y \in x$.

We need a "relativizing" predicate $U(x)$, meaning : $x$ is an individual (or atom).

Thus we have the definition [page 24] of "collection of $\Phi$'s", for a first-order formula $\Phi(x)$ :

the term $\iota ! y(\lnot U(y) \land (\forall x)( x \in y \Leftrightarrow \Phi(x)))$ is abbreviated $\{ x : \Phi(x) \}$ and we read it as "the collection of all $x$ such that $\Phi(x)$" (if it exists).

Then we have [page 30] the definition :

we say that $b$ is a collection if $b = \{ x : x \in b \}$.

We have to "unwind" the defining formula, noting that, in this case, the f-o formula $\Phi(x)$ is $x \in b$ :

$b = \iota ! y(\lnot U(y) \land (\forall x)( x \in b \Leftrightarrow x \in b))$.

But $\vdash (\forall x)( x \in b \Leftrightarrow x \in b)$ and thus, by logic alone, we have that the seemingly "circular" definition simply amounts to saying that:

$b$ is a collection iff $b = \iota ! y(\lnot U(y))$

i.e. : $b$ is a collection if it is not an individual, as you can see from the following :

(2.6.1) Lemma. No collection is an individual.