Firstly note that a set construction involves the assigning of a reference

A = {1,2} = {x : 0 < x <3}

The character A, the set name, is not usually part of the symbols that build the set. If however we did used the symbols {1,2} as a name then the statement

{1,2} = {1,2}

does not mean that the set {x : 0 < x <3} contains the number 1 and 2, but rather that {1,2} is a name that - meaningless in itself - will now refer to the set {x : 0 < x <3}. It could be expressed like this for clarity:

"{1,2}" = {1,2}

1.0) Now consider a universal set containing three sets U = {A,B,C} where,

A = {10}
B = {1,2}
C = {3,4}

2.0) Now construct the set D = {x : x = 2} i.e. a set whose members are sets with 2 members, and call it D

2.1) D = {B,C}

But D has two members. If D is not included in itself then D is not the set of "all" sets with 2 members because it is the exception. We are thus inclined to include D to give the set

2.2) D = {B,C,D}

However now D has 3 members and so should not be included. How ever we try to build D we end up with a contradiction. Where is the problem?

Up until step 2.1 everything was good. It was only "after" constructing D that we noticed it became an exception to its own rule.

Step 2.1 suggests statement 2.2 but something goes wrong in building it. Let us recreate the process without assigning any names or references (beyond the basic elements i.e. 10,1,2,3,4}

supposing we write statement 2.1 as {{1,2},{3,4}}

We notice that it contains 2 members and so should be a member of itself so we try to write the self-referential statement without using reference:

{{1,2},{3,4}, itself} = {{1,2},{3,4}, {{1,2},{3,4}, {{1,2},{3,4}, {{1,2},{3,4}, itself}}}}

it cannot be done. This expression cannot be resolved into "definite" sets. The process of attributing a reference D to the unresolved set in 2.2 hides the fact that we don't yet know for sure what the set is. Writing D looks like a definite answer but actually it is an "unbound" reference. By "unbound" I mean we is unsure what it refers to.

The statement D = {B,C,D} is a formula for the answer. B and C are bound references because we know clearly what they are, but exactly what D refers to is uncertain. When we ask what D refers to we get another formula containing D. It resolves to nothing, a trick of language without reference, meaningless. Creating formulas is easy. g = 4*Hg¬K but until we know what these refer to it is meaningless.

"The unicorn in my garden is eating grass" is meaningless because "the unicorn" is an unbound reference, it is a formula we might use in the event of a unicorn being in the garden. When asked "which unicorn?" we cannot simply answer the unicorn eating grass (q.v. what is D it is {B,C,D}).

Predicate logic resolved this issue with: Ex: x = unicorn & x = eating grass. It is false because there is no unicorn (and it certainly then can't be eating grass) but the issue of reference is implicit.

Quick addition on reference. Reference is not the linking of "physical reality" to symbols and words, because objects in the world can only be referred to with symbols i.e. what do we mean by "physical reality" since that itself is a reference! mystery!

So we are stuck with D = {B,C} but it is unsatisfactory because it becomes an exception to its own rule.

The solution might be that D = {x : x = 2} creates a new Universal. Remember that U = {A,B,C} so the construction of D creates a universal U* = {A,B,C,D}

D = {x : x e U & x = 2}

And that is the point about self-reference. The reference to oneself cannot be in the same universe as onself.

It is the +1 conjecture again that a system needs to expand by 1 to take a useless entity and use it to reference the self.

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Can we get from this position to the general situation that the self cannot contain itself.

And is this the same as the situation where the rules of a game (like war) can only occur because they try to reference themselves within themselves i.e. we Allies without understanding the requirement to see the Enemy and the Allies are part of the same game.