Confusion on definition of predicate

Question

I'm currently reading up on Predicates(Logic) and have come to understand that predicates are a finite set of variables that become propositions when the variable(s) are substituted in with appropriate values.

I'm then given an example of predicate:

(Goldbach’s Conjecture): Every even number larger than 2 is the sum of two primes.

It was then formally rewritten to:

For all n belonging to {even integers ≥ 4}, there exist p, q belonging to {primes} such that n = p + q.

1st part of question:

How is the set {even integers ≥ 4} considered finite(which according to definition above, it must be)?

The author then proceeds to say this:

To be propositions, predicates need to have some quantifiers and relations that specify what the variables are allowed to be

and this(he gave some examples of predicates before writing the below statement):

...But these statements have something in common other than the quantifiers: they all contain relations specifying the set where the variables are allowed to live

So, a term known as "relations" was suddenly introduced which is said to specify the domain of variables and is part of every predicate?

2nd part of question:

Does this mean that the above definition of predicate is lacking and must include the idea of "relations"?

Source of material: https://web.williams.edu/Mathematics/lg5/200/PropPred.pdf

Answer 1

There are only three variables there, $n,p,q$. That is a finite count.

The Integers is the source for the values for the one variable $n$.

Likewise the Primes is the source for the values for the two variables $p, q$.

So you do have a (countable) infinite source of permissible values these three variables may take (their "Domain" to be technical), but there are still only three variables.

Answer 2

Goldbach’s Conjecture is the statement:

"Every even number larger than 2 is the sum of two primes."

We can express it using two predicates: "to be a number larger than 2" and "to be the sum of two primes."

We can rewrite it in symbolic form using predicate logic.

The language of first-order logic uses individual variables: $x,y,\ldots$ and predicate symbols: $P,Q,\ldots$ where each predicate symbol formalizes a property (unary predicate symbol) or relation (binary, ternary, etc.).

In addition, we have to use the propositional connectives and the quantifiers.

Thus, we define $P(n)$ as "$n$ is a number such that $n > 2$" and $Q(n)$ as "there are two primes $p$ and $q$ such that $n=p+q$".

In symbols (assuming the domain of natural numbers): $P(n) := n > 2$ and $Q(n) := ∃p∃q \ (\text {Prime}(p) ∧ \text {Prime}(q) ∧ n=p+q)$.

As you can see, in $Q(n)$ variables $p,q$ are quantified and the only free variable is $n$.

Thus, both $P(n)$ and $Q(n)$ are unary predicates, expressing properties of the number $n$: in other terms, they depend on a finite set of variables (only one) and they become propositions when the variable(s) are replaced with appropriate values.

In conclusion, Goldbach’s Conjecture is formalized with the following statement:

$∀n \ [n>2 → ∃p∃q \ (\text {Prime}(p) ∧ \text {Prime}(q) ∧ n=p+q)]$.

According to the author's assertion, it is a proposition (a declarative sentence) because the predicates are quantified.

Finally, regarding the statement: "the predicates contain relations specifying the set where the variables are allowed to live", I've assumed implicitly that the domain is the set of naturals.

We can instead be explicit about it writing :

$\forall n [n \in \mathbb N \land n > 2 \to \ldots ]$

and the same for $p,q$.

Note: if we omit the leading quantifiers in the definition of $Q(n)$ what we get is the formula $(\text {Prime}(p) ∧ \text {Prime}(q) ∧ n=p+q)$ expressing a ternary relation $R(n,p,q)$ between numbers: $n,p,q$.