# Subset of Theoretical Practice

## 20 questions

One can conceive of the phenomenological object as a space for the game "20
questions"^{[1]}. In this game, one player (let us call her the oracle) privately
commits to some ontological entity (typically restricted to an agreed-upon
type), while the other asks a series of yes or no questions, up to 20, to find
the entity. Each question determines a set: "is it man-made?" yields the set of
"man-made" things. If the answer to a question is yes, the entity in question
lies in the set, and if the answer is no, the entity lies in the complement of
the set. The set of questions leading to an answer constructs what
mathematicians call a "filter" (c.f. "net"), the collection of subsets
containing some element, which can be ordered by inclusion (c.f. "restriction").
The space of such a game consists of all ontological entities of a certain type,
along with a certain topology to be discovered by both players. To win against
the oracle, one must choose those questions which together most approximate a
filter (of height ≤ 20) for your target. In logical terms, the predicate π which
uniquely singles out the entity is a conjunction of predicates π_1, π_2, ... π_n
(where n ≤ 20). The value of each question then is determined by how much it
excludes, i.e., the size of its complement-set - which is in turn determined by
the questions and answers that came before it. In some cases, questions may not
yield a true or false answer, but nevertheless yield information. Consider the
entity "human" and the question "is it man-made?". Or "virus" and "is it
alive?". Or "silence" and "is it part of language?". The underlying premise is
that every being of a particular world has an address of at most 20 predicates.
Likewise, to win as the oracle, one should commit to an entity that is not
addressable, that most resists any anticipated line of questions, something
which might be "virtual" with respect to the type of world agreed upon by the
two players. To be fair, the world should be restricted to no more than 2^20
entities. If the domain is that of rational numbers for example, the oracle
should have no problem winning. But even in a finite world, one cannot prevent
diagonalization by a skilled oracle: "the being predicated on the negation
of every predicate in the space". Conversely, every
entity in the world can be said to generate its filters. Just as the conjunction
of predicates produces a more precise answer,
their disjunction will eventually lead to the entirety of the world. But there
are multiple paths to generalization. One can form a "universal" way of getting to the answer from the particular filters, i.e. by collecting the set of all predicates from all
filters based on the entity. We call this an "ultrafilter". With finite
sets, every ultrafilter is also "principal", meaning it contains a least
element - in our case, the object of the game. With infinite sets, things are not so simple ^{[2]}. Namely, it is possible to prove the existence of an ultrafilter without reference to an element generating it. This leads to special kinds of spaces that are neither compact nor Hausdorff. In topology, spaces in which
every ultrafilter converges to at least one point are "compact", and if they converge to at most one point,"Hausdorff" ^{[3]}. And in
model theory, the process of adding a certain ultrafilter based off of "fictional" points
is what lies at the heart of creatively extending a model^{[4]}. In a sense which
can be made infinitely more precise, we can play the game in reverse: we can
produce a new world by adjoining something alien to the existing one. The game
of 20 questions is therefore a toy version of the game of countably-infinite
questions, that is, mathematics (ontology) itself. The productive interplay
between analytic specification and synthetic generation cuts across Badiou's
project. It informs every one of the formal choices that he makes. In light of
this, we can reconsider the following from Theory of the Subject ^{[5]}

"We welcome those vicinities of the vague in which the partitive multiplicity is dissolved, considering them to be the proof, administered by those who would desire the exact opposite, that there is a wager on the real. If, in this wager, the number inscribed on the dice is the result of a consecution, it cannot link up into a chain that which, in the thrower's gesture, produces the incalculability of its reach."

"20 Questions", https://en.wikipedia.org/wiki/Twenty_Questions ↩︎

"Ultrafilter (set theory)", https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)#Free_or_principal ↩︎

"Compact Hausdorff Space", https://topospaces.subwiki.org/wiki/Compact_Hausdorff_space ↩︎

In the extreme case, these are the generic filters. See: https://en.wikipedia.org/wiki/Generic_filter ↩︎

Badiou, A. (2009) Theory of the Subject, Continuum. p. 274 ↩︎