{∅, {∅}, {∅, {∅}}}

implies { }

The singleton of the void is written here as {∅}.

The name of the void is written here as ∅, meaning the empty set, { }, NULL.

{∅} is the formation-into-one of the name of the void. Its sole element is ∅.

What would the parts of the power set of p(∅) be?

There is {∅} itself; there is also ∅ because the void is universally included in every multiple.

∅ is part of every set. The multiple p(∅) has two elements, ∅ and {∅}.

Here, woven from nothing apart from the void, we have the ontological schema of the Two, which can be written {∅, {∅}}.

The element ∅ is part of the Two.

The element {∅} is also part since ∅ is an element of the Two (it belongs to it).

The two elements of the two are also two parts of the two. The mathematical concept of transivity is therefore possible. Transitivity tells us, “everything which belongs is included.”

The inverse is impossible.

It is not possible for everything which is included to belong.

Not only is the Two a transitive set, but its elements, ∅ and {∅} are also transitive.

Nothing inside ∅ is not “a part” since the void is no thing, the empty set, null.

 

A Few Comments About ∅

As stated, ∅ is a symbol for the empty set, { }, “the void” —-> NULL.

The void is a subset of any set: it is universally included.

The void possesses a subset, which is the void itself.

Examining these properties of the void is an ontological exercise.

The first property testifies to the omnipresence of the void.

The void, to which nothing belongs, is by this very fact included in everything!

The State considers the individual as a subset – not as Your-Government-Name (the proper name of an infinite multiple) but as {Your-Government-Name}, an indifferent figure of unicity, constituted by the forming-into-one of the name.   The individual is included within the State.

Coercion consists in not being held to be someone who belongs to society, but as someone who is included within society.

Does this make sense?  Note the difference between BELONGING to society and being INCLUDED within society.  A stateless person doesn’t “belong to” any state.  A de facto stateless person is someone who is outside the country of his or her nationality and is unable or, for valid reasons, unwilling to avail himself or herself of the protection of that country.

The relation of belonging is the fundamental non-logical relation that structures all sets, and is written ‘∈’.    The relation of being included is written ‘⊂’.  I’m not sure if the difference is clear to me.

Given a set A = {a, b, c, d}, the elements that belong to it are: a, b, c and d.

But what about sets that share coincident elements, such as B = {a, b} for example?

Such a set is said to be included in A, or to be a subset of A, and is written: B⊂A.

If all the elements of a set B are also elements of A, then B is included in A.

The Power Set Axiom then states that if a set A exists then so does the set of all A’s subsets.

Taking the example C = {a, b, c}, the power set of C is: p(C) = {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, ∅}.

The new set, p(C), has eight, or (2 to the 3rd power) (2^3) elements. Perhaps the only two surprising inclusions are the empty set and the set C itself. Given the definition of a subset, their inclusion becomes clear. Although the original set cannot belong to itself, on pain of paradox and inconsistency, it can include itself as it obviously shares all its elements.

The empty set, ∅, has the unique property of being universally included in all sets; there is no element belonging to ∅, which is not also an element of any other set, as ∅ has no elements.

If sets present their elements, they represent their subsets. The full representation of a set is equivalent to its power set, and Badiou calls this the State of a situation. The State represents the situation, and it will be in the minimal relation between an infinite set/situation and its power set/State that novelty will be possible.

Badiou’s set theoretical universe is sparse; only the empty set exists.

The first new set he produces is p(∅) = {∅}, a set with one element, a singleton.

This is not too surprising either, if the general rule is that the number of elements of a power set are 2^n, where n is the original number of elements, if n = 0 then 2^0 = 1.

From this Badiou derives the rule that given any set A, then its singleton, {A}, also exists.

We are now in a position to consider the construction of the finite ordinals. The void, or empty set, ∅, can be considered as the first natural ordinal 0, with its singleton {∅} corresponding to the ordinal 1.

The successor of these two ordinals is the union of these two: ∅∪{∅} = {∅, {∅}}, the ordinal 2.

The process of succession is to form the unity between the current ordinal and the singleton of this current ordinal. The construction of the ordinal 3 is accomplished as follows: the union of {∅, {∅}} with its singleton {{∅, {∅}}}: {∅, {∅}}∪{{∅, {∅}}} = {∅, {∅}, {∅, {∅}}}. In general if a is an ordinal the successor of a is a∪{a}.  This is equivalent to the idea of adding one. The interesting feature of this construction of the ordinals is that all the previous stages of the construction appear within the current level as elements. Every element of an ordinal is itself an ordinal.  It is this feature of nesting and homogeneity that qualifies a set as transitive:

∀A∀B (A∈C & B∈A) → B∈C

This reads, if A belongs to C and B belongs to A, then B belongs to C.

Now we go insane.

{∅}

{∅, {∅}}

{∅, {∅}, {∅, {∅}}}

{∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}

{∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}, {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}}

 

~ H : {∅, {∅}, {∅, {∅}}}

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