The Transcendental Set

Propositon: a set cannot be a member of itself

Using c to mean is a subset of, ¬c to mean not a subset of, e to mean is a member of, and ¬e mean not a member of, n to mean intersection

Take a universal set (U) of natural numbers: N
If E is the set of all the even numbers, then E c N but E ¬e N because "the set of even numbers" is not itself a number. (i.e. A is a subset of N, but not a member of N.)

Thus the process of "creating" a set seems to be an arbitrary "given" action performed by the thinker which is not present within the theory of sets. In other words, you cannot express the process of set creation within set theory, it is a mysterious process of "mind" that is required prior to set theory itself.

We can create a new universal set: the set of all natural numbers and sets of these numbers = N1

Now E c N1 and E e N1.

But if we create a new exotic set say all the even numbers and itself, E1

Then again E1 c N1 but E ¬e N1

Now we can continue this recursion one stage further, N2 = the set of all natural numbers, sets of these numbers and sets of these sets. There is an infinite recursion at each stage the new Nx not being a member of itself.

Is it possible to close this infinite recursion by being self referential? So that T was a set of sets of such sets to an infinitely level of recursion i.e. T e T

We would need to say something like: T = the set of all sets where each member x, if x ¬e N can be unpacked recursively until x e N.

However there is the obvious problem that this loop cannot be resolved because T = {T, ...} and at each stage it must start again to unpack itself.

However if we were to create a super set without complex conditions basically the set of everything. A set that truely can belong to itself. I have come to consider what the meaning of such a set is? Can it actually be formulated, is it meaningful?

Consider so far: there is the arbitrary process required by the human operator to construct a set. Even just using the empty set {} an infinite set can be created as is used to define the natural numbers (http://plato.stanford.edu/entries/set-theory/primer.html/l4). Quite how we construct sets tho is quite arbitrary and requires the "intelligence" of an operator to "decide" what to do. Logical systems can't solve themselves else the whole of maths and thought would be there on a plate. So while we can consider such a set, real set theory requires meaningful decisions by the operator to "create" sets and monitor the process of proof and deduction. The set T is composed from an indefinite number of meaningless and contradictory sets (like the set of all sets that do not contain themselves). It is a real chimera of no rigorous value.

Let us take such a transcendental set T which has one member namely itself.

T = {T}

This defines a recursive set so that unpacking any number of times gives us equalities of the form:

T = {..{{{{T}}}}..}

If we try to create a new set containing the transcendental set {T} we find that it is the same as T. The recursion works indefinitely in both directions (i.e. unpacking and repacking).

T has only one member itself, but defines an infinite "heirachy" of sets. Heirachy was the property that stopped us previously making {A} a member of A.

Now let us take an exotic transcendental set with a second non transcendental member.
T = {T, A}

This creates a recursion downwards when we try to "unpack" and a heirachy of the form
T = {.. { { { {T, A}, A} ,A} ,A} .. ,A}

However If we repack (recurse the other way) we can ask what is {T} ? i.e. what is the set containing T?

It clearly isn't T itself (which is {T, A} ) and it isn't a member of T either.

Again the process of heirachy stops us from including T within itself. It is no-longer a transcendental set.

If however we take the set A again (which is a subset of T) then we could repack the two together {T, A} and continue the recursion backwards.

...this is to be completed...

I was also thinking that the first exploration shows us that when a set is determined by reference to members it can't be a member of itself because there is the recursive problem of finding out what the members are.

However if a set is defined by reference to set properties that are clear regardless of the members then we can determine set membership without recursion, and then the possibility of self inclusion arises.

To investigate that I was considering the properties of a set. Let P be the set of all set properties
P = {definition, description, cardinality}

P should be self referential at least in that if it is a set it should fulfill those criteria.

P = {"a list of different entities separated by commas and enclosed in brackets", "the list of all set properties", 3}

Now using these we can construct sets without reference to their members, and so avoid the recursion problem.

1) A set containing a set: T = {T}

2) A set whose member is named T: T = {T}

3) A set containing 1 member T = {T}

But these are rather fake examples. A necessary example would be the set of all sets which MUST contain itself since it is a set. T = {T, ...}

I'll leave the investigation there for now till its clear where to go...