The True Nature of the Indefinite: The Part for the Whole

Then if $a$ is different from $b$, then $j$ is different from $i$. In simplified language, it is said that a set $A$ is indefinite if there exists an injective function $P: A \to I$ where $I$ is a part of $A$ that does not coincide with $A$. It is in $I$ being contained in $A$ but being able to accommodate, through a procedure $P$, a faithful image of $A$, that the nature of $A$ being indefinite is revealed. If $A$ were not indefinite, every faithful image of it could not be entirely accommodated in any of the parts of $A$. This understanding of the indefinite as that which admits a part that is a perfectly faithful image of it lends itself to supports for meditation that are very important also for understanding the intermediate world or errant influences. Perhaps this indirectly clarifies the reluctance to explicate many aspects of the symbolism of the indefinite. The true nature of the indefinite: the part for the whole. Let us meditate well on what the existence of a part $I$ contained in an indefinite $A$ entails, which satisfies the condition of existence of an injective function $P: A \to I$, that is, which satisfies the two conditions (1) and (2) described above. An observation we can immediately make is that if we assume that the famous principle of the excluded middle holds, then we can, in a certain sense, invert the procedure $P: A \to I$ and through it construct a function, that is a procedure, $Q: I \to A$ that reproduces all of $A$, or rather such that for every element $a$ of $A$ there exists an element $i$ of $I$ such that $i \xrightarrow{Q} a$. Let us stop the flow of words. We are seeing how to reconstruct all of $A$ starting from one of its parts exactly as from the halving procedure we reconstruct all numbers starting from only the even numbers. In this case the procedure $Q$ was given, for example, by the halving operation. Actually, in the case of any indefinite, an analogous procedure always exists, provided that the principle of the excluded middle is admitted. Indeed, let us concentrate the mind on an element $a_0$ of $A$ and isolate it from the rest of the elements of $A$. Now let $a$ be any element of $A$ and let $i$ be any element of $I$. Since in this meditation we can use the principle of the excluded middle, then we can verify whether for every $i$ of $I$ there exists or does not exist an $a$ of $A$ such that $a \xrightarrow{P} i$. In other words: being able to use the principle of the excluded middle, we can immediately see if an element $i$ of $I$ is or is not (it is in this either-or that the principle of the excluded middle manifests itself) in the image of $A$ in $I$ obtained through the procedure $P$. We can therefore define a sort of inversion of $P$ in the following way $Q: I \to A$ where: $i \xrightarrow{Q} a$ if and only if $a \xrightarrow{P} i$, and otherwise $i \xrightarrow{Q} a_0$. Since we can use the principle of the excluded middle in the case of an indefinite, the part $I$ dominates over the whole $A$. A function like $Q: I \to A$ such that for every $a \in A$ there exists $i \in I$ such that $i \xrightarrow{Q} a$ is called a surjective function. It could be demonstrated that if for every indefinite $A$ and if for every part $I$ of $A$ whenever it happens that there exists an injective function $P: A \to I$ then there exists a surjective function $Q: I \to A$ is completely equivalent to the principle of the excluded middle. Actually we see how the analogy between the case of the subpart of even numbers with respect to all numbers and the case of the indefinite is exactly the same: in other words if $I$ is a subpart of $A$ such that there exists an injective function $P: A \to I$ then there exists a bijective function $f: I \to A$, that is a procedure such that given $a \in A$ there exists one and only one $i$ of $I$ such that $i \xrightarrow{f} a$: in the case of the indefinite there is at least one of its subparts, $I$, that stands for the whole. This is the content of a famous theorem known as the Cantor-Schröder-Bernstein Theorem. We care here, first, to observe that the existence of a bijective function $f: I \to A$ is a way to overcome a purely quantitative approach to meditate on certain numerical realities. The existence of a procedure $f: I \to A$ that to each element $a$ of $A$ makes correspond one and only one element $i$ of $I$ shows that we can look distinctively at all the elements of $a$ by looking, distinctively, first at the elements of $I$ and then applying to them the procedure $f$ and in this sense the vision that $I$ has as many elements as $A$ has jumps to mind. We do not really answer the question "how many are the elements of $A$" but we distribute reality according to the relation "as many as". The correspondences between entities $A$ and $I$ that have the property of being extensively bijective, that is, that are established by bijective functions $f: I \to A$ are not arbitrary acts of reason but acts sometimes very useful to be able to meditate effectively on Reality. We deem it useful to comment on a proof of the celebrated theorem of...