What is Tarski’s definition of real number multiplication?
The results of our searching for Tarksi's definition of multiplication is given in the next section. In this section we 'cut-to-the-chase', sketching how to apply the Eudoxus theory of ratios.
Like Eudoxus/Euclid and other ancients, in this exposition numbers will always be positive; we are working in $(\Bbb R^{\gt 0}, 1, +)$. Before starting note that $(\Bbb N^{\gt 0}, 1, +)$ is naturally included in $(\Bbb R^{\gt 0}, 1, +)$.
We define the ratio of $u,v \in \Bbb R^{\gt 0}$ as a binary relation in $\Bbb N^{\gt 0} \times \Bbb N^{\gt 0} $,
$$\quad u \mathbin{:} v = \{ (n, m) : nu > mv\}$$
where $nu$ and $mv$ represent repeated addition. So the ancients could work with real numbers via ratios without a decimal system.
As a sanity check, the power set of $\Bbb N^{\gt 0} \times \Bbb N^{\gt 0} $ has the power of the continuum.
We only state what we need here from the ancient theory of proportions
(c.f. Euclid's Elements.Book V.Proposition 14).
Theorem: For any $x, y \in \Bbb R^{\gt 0}$ there exist one and only one number $z \in \Bbb R^{\gt 0}$ satisfying the following
$\quad \text{For every } m, n \in \Bbb N^{\gt 0}$
$\quad \quad \quad \quad \quad \quad [$
$\tag 1 nx \lt m \; \text{ iff } \; nz \lt my$
$\quad \quad \quad \quad \quad \quad \text{and}$
$\tag 2 nx = m \; \text{ iff } \; nz = my$
$\quad \quad \quad \quad \quad \quad \text{and}$
$\tag 3 nx \gt m \; \text{ iff } \; nz \gt my$
$\quad \quad \quad \quad \quad \quad ]$
You can think of above result as a variation of the 'squeeze theorem' by letting $n \to +\infty$ and taking the largest $m$ such that
$$\quad \frac{m}{n} \le x \le \frac{m+1}{n} \; \text{ and } \; \frac{m}{n}y \le z \le \frac{m+1}{n}y$$
is true.
Definition: For any $x, y \in \Bbb R^{\gt 0}$ the number $z$ from the theorem is denoted by $x \times y$. The corresponding binary operation on $\Bbb R^{\gt 0} \times \Bbb R^{\gt 0}$ is called multiplication.
I found an online version of Tarski's book.
The book DOES NOT define multiplication!
In the last chapter of the book, chapter 10, two axioms systems for the real numbers are presented in a 'survey' fashion,
$\mathcal A'$ (the one where the OP lists the axioms) and in summary Tarski writes
System $\mathcal A'$ expresses the fact that the set of all numbers is a densely and continuously ordered Abelian group with respect to the relation < and the operation of addition, and it singles out a certain positive element 1 in that set.
and
$\mathcal A''$, and in summary Tarski writes
System $\mathcal A''$ expresses the fact that the set of all numbers is a continuously ordered field with respect to the relation < and the operations of addition and multiplication, and singles out two distinct elements 0 and 1 in that set, of which the first is the identity element for addition, and the second, the identity element for multiplication.
In Section 62
$\quad$Closer characterization of the first axiom system;
$\quad$its methodological advantages and didactic disadvantages
Tarski writes that
Even constructing the definition of multiplication and deriving the basic laws for this operation are not easy tasks to carry through.
Later in Section 65
$\quad$Equipollence of the two axiom systems;
$\quad$methodological disadvantages and didactic advantages of the second system
Tarski writes that
...both the definition of multiplication on the basis of the first system, and the subsequent proofs of the basic laws governing this operation, present considerable difficulties (while these laws appear as axioms in the second system).
Now wikipedia writes, in regard to $\mathcal A'$,
Tarski sketched the (nontrivial) proof of how these axioms and primitives imply the existence of a binary operation called multiplication and having the expected properties, so that R is a complete ordered field under addition and multiplication.
But that sketch is not to be found in the one (relevant) reference wikipedia gives - the book the OP is reading!
Wikipedia also states
Tarski proved these 8 axioms and 4 primitive notions independent.
And again, no reference.
Besides the primitive terms, $\mathcal A''$, with multiplication 'built-in' contains 20 axioms.
The last thing you will find in the book (besides the index) are the exercises for Chapter 10 and the last exercise is
*22. Derive all the axioms of System $\mathcal A'$ from the axioms of System $\mathcal A''$.
Tarski's book doesn't have any references.
The OP might find the link
Talk:Tarski's axiomatization of the reals
of interest. Apparently some mathematicians are trying to come up with the definition of multiplication in System $\mathcal A'$, and one came up with
