The Zero Delusion v0.99x

with the expression
{1} , {1, {1}} , {1, {1, {1}}} , {1, {1, {1, {1}}}} , · · · .
The former considers {} a set, i.e., presumes that "nothing" is a countable
thing, and adopts von Neumann’s interpretation that every set has as elements
the sets with ordinality lesser than it so that ordinality and cardinality coincide.
The latter eludes the empty set and is a Zermelo-wise interpretation where every
set has two elements, i.e., 1 and the previous set. In this case, the ordinality of a
set corresponds to the sum of ones or the nesting depth ascribed to the last item,
e.g., Ord {1, {1, {1, {1}}}} = 4. This account weds arithmetic to set theory more
straightforwardly and is a neater semantic of individuating the natural numbers
rooted in thought and language [113].
Moreover, all properties of sets are trivially valid for the empty set; for instance,
the Axiom of Union and the Schema of Replacement claim nothing substantive.
Valuable theorems refer without exception to nonempty sets, and even
many axioms and theorems of ZFC specifically exclude the empty set, e.g., the
Axiom of Regularity and the Axiom of Choice. What binary relation well-orders
the empty set? None. What is the service of the intersection or Cartesian product
of a set with the null class? None.
Users of the ST inherently assume the convenience of counting on, for example,
a crate, bag, or trashcan vacated. Nevertheless, the point is that ST refers
to "the" empty set. Thus, finding two empty baskets is inconvenient, a situation
against the theory. What is the effectiveness of a bin, lone, doomed forever to
hold nothing? There are infinitely many sets of any cardinality, with the sole exception
of the empty set. The only set from which we cannot remove an element
and the only one that is a (strict) subset of any set except itself! If the definition
of subset requires elements in common, the statement {} ⊂ S is false, whereas
if it requires that no elements in the contained set are out of the container, the
empty set is a subset of itself. Moreso, the empty set implies "vacuously true"
statements that give rise to paradoxes, e.g., ∀x {} → x (anything is provable).
Worse yet, it is a topological self-contradiction; despite representing a hole, i.e.,
a no-point, the empty set, closed and open simultaneously, has the property of
"compactness" (neither punctures nor missing endpoints). Given these facts, we
daresay that the empty set is useless, weird, or even an encumbrance.
Handling only populated sets means wriggling away from this exceptionality
and gibberish. Thus, a set proved to be empty should formally disappear, an
approach that agrees with Hausdorff’s view when he wrote that A = {} means
that A vanishes (section 2.1 in [69]). If we needed intersections of sets even when
there is no intersection, we could take such an intersection to be S ∧ ¬S, again
a dull vanishing expression. We can handle "copies of the empty set" as beables
to cope with collections possibly deprived of content. For instance, consider a
handful of marbles distributed in bags A and B, with A ⊂ B. How many of them
are in B − A? We must take A as a beable, a not-being yet but a potential set.
If B − A ≠ B, A is a set; otherwise, A is empty, so we rule it out as a set, and
our new universe is exclusively B. An empty beable is equivalent to handling a
future set. For example, one may prefer to keep a bank account with no money