I want to state that I agree with critisism raised by other replies. Especially that the definitions are causing the problem.
However I have further criticism that haven't been voiced yet.
Let the world (w) be defined as: the domain that contains all domains.
Let existence be defined as appearing within a domain. — auto to on
1) "The world" seems to imply that there is only one world. Therefore the introducing of multiple worlds (w1,..., wn) is unjustified since they contradict the first definition.
2)The definition of existence presupposes the existence of a domain in which elements can appear in. Therefore if we claim the world exists we either presuppose a different kind of existence (existence2) of the world in order to ascribe existence1 (existence as used in the definition) to the world. Or we get a circular reasoning pattern where the existence of the world is a prerequist for existence, and existence is required for the existence of the world.
Furthermore if we claim that the world exists we don't get w2 that is within w1 since we only ascribe existence to w2. Therefore w1 would be the worlds pseudoexistence (existence2) and w2 would be the world with ascribed existence1. However they both refer to the same world w.
I know that existence is not a property and rather a quantifier but since you use existence rather confusingly I'll describe it as if existence was a property. If we have an object O and we ascribe two different properties to O we don't get O1 and O2 and rather O with properties P1 and P2.
Therefore the confusion most likley arises from the questionable use of the term existence and it's two forms that are not distinguished properly. If we see this as faulty or insufficient description we get a faulty tautology because of the definitions used.
This means that any other deduction based on this definition is flawed. Therefore viewing any other transformation in this tautology as consistent because of conclusions that might be true is a logical error. (From False everything follows)
This applies to your response stating:
Other conceptions may be consistent with these definitions — auto to on
3) First of let's note that S1,...,Sn just appear randomly and are not defined and irrelevant to the problem. Therefore we can leave them aside.
Note: I am aware that you think theres a problem if S1 appears alongside w1 but also within w1 however theres no definition given that states something among the lines of "if something appears alongside x this something can not be within x". You generally seem to establish rules that are not given by the definitions. Another example would be something containing itself creates a new instance of itself. This is needed to even argue that you get w1 and w2. (It's like assuming that if I say I am my best friend that you assume theres two of me, since you have in your head that the best friend is a different object despite no rule stating that.)
Since you somehow want to hold on to your definition for no clear reason lets adress that aswell.
Let us therefore assume that the world (w1) [...] is contained within world (w2) [...]. Now we are met with the same difficulty, if the world properly contains itself then this should obtain (w1) = (w2). [...] but it[(w1)] must also contain itself, let us call the world contained within (w1) for (w0). — auto to on
What you are doing here is create a problem where there is none. Since w1=w2 and w2 contains w1, w1 contains w1. Or in other words w1 contains itself.
We can show this aswell if we accept that w0 is the world contained in w1 and show that w0 = w2.
This can be done intuitivly by saying that w0 is the domain that contains all domains and that the same is true for w2 and that there can only be one domain of all domains.
We could also state the reason given in 1) and conclude that you are just making up new names for the world and think this is a logical problem.
However lets assume the premisses you defined as trivially given for w2 (our initially defined world).
Since (w) in (wn) refers to world and world is defined as domain of all domains (wn) trivially is a domain (d).
Further every (w) in (wn) refers to world and world is defined as the domain of all domains (D)
Since every domain (d) is contained(<=) in the domain of all domains (D)
It follows that (d) w2 <= w0 (D)
Since you have already established that w0<=w1<=w2 and therefore w0<=w2
we get w0=w2 or more general wj=wi where i and j are arbitrary numbers. We would also get w1=w1 or in other words w1 contains itself.
In conclusion your set of definitions is faulty or at best incomplete. Even if we ignore the flaws the problem you make up doesn't exist since you just rename a single instance and think the ability to establish an infimit ammount of names equates to a infinit regression since you presume that ascribing a new name creates a new object.
I hope I understood you correctly and that this helps solving your problem.^^