I some small questions, I am now learning about the interpretation of the language $L_0 L_1$ which has these symbols: $(A_2^2, A_1^2, f_2^2, f_1^2, a_2, a_1)$ and can describe this set: $(\mathbb{Z},0,1,+, \cdot,<)$
$f_1^2$ - is the addition $f_2^2$ -is the multiplication $a_1 =0, a_2 = 1$ and $A_1^2$ is the equality predicate ($A_2^2$ is the less than predicate)
In the book they said that $A_2^2$ ($<$) is defined as a relation
$$P_2^2: \mathbb{Z}^2 \rightarrow \{T,F\} \\ P_2^2(d_1, d_2) = (T ~~ \text{if}~~ d_1 < d_2), (F ~~ \text{if} ~~ d_1 \geq d_2 ) $$
But isn't it a recursive definition? they literally defined $<$ using $<$ ... I don't get it..? If they say "let's define what is $<$ then we shouldn't use this as a symbol anywhere in our definition because it is not defined yet! in addition, what is $\geq$ ? I mean, I know what it is of-course, but - what is it in this context? It is not even a predicate in our system! $(\mathbb{Z},0,1,+, \cdot,<)$ so how can we still use this? we have not defined this "object" $\geq$ ...
Last question, I thought that a predicate is the whole "mathematical text" as such: $\forall x_1 ( \alpha)$ or $\exists x_n( \alpha \rightarrow \beta) $ for example... but - now they call $=$ and $<$ predicates, so, what is really a predicate? is it just these symbols $=$ and $<$ ? only the ones that give $\{T,F\}$ ? Not the whole text itself? Thank you!
I see that the double usage of "$<$" is confusing.
When doing predicate logic, we define a formal language. A formal language of predicate logic contains non-logical symbols: constant symbols (= symbols that stand for individual objects), predicate symbols (= symbols that stand for properties of and relations between objects), and function symbols (= symbols that stand for functions on objects).
In mathematical practice, it is convenient to choose the non-logical symbols such that they convey the intended meaning according to which we want to interpret them. If we want to formally talk about the "smaller than" relation, it makes sense to choose the symbol $<$ rather than just something like $P$ or ☺, in order to make the formulas more readable.
But what is the "intended meaning" still has to be explicitly defined and thereby fixed "from the outside". Logic "doesn't know" that the predicate symbol $<$ is supposed to mean "smaller than". In principle, there is absolutely nothing that prevents it from meaning "is greater than", or "is a less beautiful number than". By itself, $<$ when used in a formula is just a completely meaningless string.
It is the role of a structure such as $(\mathbb{Z},0,1,+, \cdot,<)$ to give a meaning to these non-logical symbols. By defining $< \ \mapsto P_2^2: \mathbb{Z}^2 \rightarrow \{T,F\} : P_2^2(d_1, d_2) = (T ~~ \text{if}~~ d_1 < d_2), (F ~~ \text{if} ~~ d_1 \geq d_2 )$, we are fixing how to interpret the symbol "$<$". The "$<$ "on the left is the predicate symbol: A previously meaningless symbol of the formal language that is now assigned a meaning. The "$<$" and "$\geq$" in "$d_1 < d_2$" and "$d_1 \geq d_2$" are "real" "smaller than"/"greater than or equal to" relations: They are mathematical objects which are already known and defined. The point of the line you cited is to explicitly state that by the formal symbol "$<$" we mean the "smaller than" relation.
The same goes for the non-logical symbols $0, 1, +, \cdot$: Logic makes no predictions on what these symbols are supposed to mean, so we have to define that we want to interpret the symbol "$0$" as the number zero etc.
As for your last question: No. A predicate is a symbol that is interpted as a property of a relation betwen individuals, e.g. $<, = \text{is-even}, \text{is-divisible-by}$. What you mean by "mathematical text" is called a formula.