Do unary relations naturally lead to the notion of sets?

Question

I have no formal background in logic, but after reading several S.E. posts about how to get started, I have begun reading Robert Wolf's "Tour Through Mathematical Logic".

One of the author's comments that has repeatedly confused me relates to unary relations. Specifically, there are several times where Wolf speaks of unary relations, and they seem to be subsequently associated with sets...or at least set-like behavior.

For example, consider the following statement, where Wolf discusses axioms that would necessarily be added to a Second Order Logic:

...if $A$ is a unary relation variable and $P(x)$ is any formula that does not contain $A$ free, then $$\exists A \forall x [ x \in A \iff P(x)]$$ is a standard second-order axiom

The sudden appearance of the binary relation $\in$ leads me to believe that $A$ is functioning as a set (as that is the only time in my minimal math exposure I have seen this symbol). The reason this confuses me is that there is never any mention of "sets" per say in Wolf's exposition of First Order Logic or Second Order Logic...which I have always interpreted as his way of describing arbitrary logic systems (i.e. not necessarily the ZFC...where sets are explicitly instantiated).

Another appearance of the $\in$ relation in conjunction with a unary relation $A$ arises in the context of one of the Peano Arithmetic axioms that Wolf describes using Second Order Logic:

$\forall A [\bar{0} \in A \land \forall n ( n \in A \rightarrow S(n)\in A] \rightarrow \forall n (n\in A)]$

This instance is evening more confusing to me because there seems to be a notion of sets here (at least to the extent that I am interpreting the $\in$ relation correctly) in a language that I did not think had anything to do with sets.

Any insight is greatly appreciated! Cheers~

Answer 1

Do unary relations naturally lead to the notion of sets?

Yes; an unary relation (or predicate) $P(x)$ expresses a property of objects.

Thus we can consider the set $A$ of all and only objects that have property $P$ and we have that

$a \in A \text { iff } P(a)$.

The introductory part of Wolf's book is quite informal: it does not states the formal specifications of first-order language

For the precise definition, you have to see page 23.

Considering the example at page 15, we may simply rewrite the restricted quantifier $(\forall x)P$ instead as $\forall x (x \in A \to P)$ as follows: $\forall x (A(x) \to P)$ and we can avoid $\in$ using, instead of set $A$, the corresponding predicate $A(x)$.

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Regarding the Second-order Logic examples, we have to consider that the only modification to FOL language is the possibility to quantify relations (and functions) symbols (see page 43).

Thus, considering the second-order logic example above and the author's explanation (see page 44), I'll suggest to re-write it using the approach described above:

if $A$ is a unary relation variable and $P(x)$ is any formula that does not contain $A$ free, then

$$\exists A \forall x[(A(x) \leftrightarrow P(x)]$$

is a standard second-order axiom.

Unfortunately, written this way we have an ambiguity. Thus, I'll prefer to use the usual schematic version of Second-order Comprehension Axiom:

$$∃A∀x[A(x) ↔ \varphi(x)]$$

where $\varphi(x)$ is a second-order formula with $x$ among its free individual variables and the second-order variable $A$ is not free in $\varphi$.