The Zero Delusion v0.99x

these are called the set’s elements or members, which can be, in turn, sets [28].
The fiducial ST’s axiomatic system is allegedly Zermelo-Fraenkel ST, along with
the axiom of choice (ZFC).
The popularity of ST is inquestionable and must be acknowledged. However,
its development took place using Predicate Calculus with the structure of the
R-line in mind, thus lacking predicativity and constructivism [40]. Alike, ST pays
exceptional attention to the Continuum Hypothesis and "actual" transfinite sets
[48,26,194]. Since many research programs and papers focus on infinite sets, we
will leave this issue out of the scope of our essay and dig into the infinity’s
mate; the complement of the universal set is the empty set "{}", which Boole
interpreted as [22] "nothing" or the class of "no beings".
Constructive theories such as Kripke-Platek’s include an Axiom of Empty
Set, stating that there exists one set with no members and cardinality nil called
"empty" or "null" [8]. Tarski-Givant’s [165] and Burgess’s Szmielew-Tarski theories
[27] also explicitly assume the empty set axiom. These theories usually formulate
this axiom as ∃S ∀y ¬ (y ∈ S), where the uniqueness of {} follows from the
Axiom of Extensionality. Still, most studied and used axiom systems do not include
the Axiom of Empty Set (e.g., ZFC ST and its variants and extensions) but
allow deriving it, for instance, through the Axiom of Specification, which states
that given any set S and a logical predicate p (x), the elements x ∈ S that make
p (x) valid form a set. Assuming that S and x exist and the Principle of Identity,
a property impossible to fulfill, e.g., p (x) = x /∈ S or p (x) = x ∈ x ∧ x /∈ x,
supposedly generates {}.
To begin with, it is disputable that the paradoxical expression {x ∈ S|x /∈ S}
(or {x /∈ S|x ∈ S}) is equivalent to {}. However, the real problem is that since
the mathematical definition of a set is extensional, all that matters is the set
content, so considering the void a class of content is counterintuitive. Indeed, {}
does not mirror the real world, where we cannot encounter empty collections,
and it is neither apparent how it can admit relations of any type. Does this
mathematical object exist? Remarkably, Dedekind excluded the empty set from
his initial essay because it was unnecessary [88], and the same Zermelo called it
"improper". According to Locke’s thought ("Substance and identity" in [114]),
"we should not uncritically accept the currently standard view that zero denotes
the empty set, becasue it is far from clear that the notion of such a set really
makes sense. [...] What is unclear is how there can be, uniquely amongst sets, a
set which has no members. We cannot conjure such an entity into existence by
mere stipulation [...]. So, [...] the reason why the property of being zero cannot
be possessed by any object or plurality of objects is that there is no such number
as zero and consequently no such property as the property of being zero."
Thus, although the null set is considered factual in practically all axiomatic
systems of ST, its existence and usefulness are suspicious. Suppose the objective
of an axiomatic system is to formalize the essential concept of a hereditary wellfounded
exclusive set ("What we are describing" in Chapter I of [103]). In that
case, one can replace (Axiom of Infinity)
· · · , {{{{}} , {}} , {{}} , {}} , {{{}} , {}} , {{}} , {}