The +1 conjecture = ambiguity problem

Firstly there is the issue of a system "actually" containing itself ... a box containing itself.

A box is a simple system with a single relevant function namely to contain things (like a set). Suppose a box could contain its real self. When would it perform this function?

Usually we put something in a box and it there by performs the containing function on that object. But if it can contain itself it forms a closed system, with self-containing an intrinsic feature. Suppose there is a condition which determines whether it performs the containing function on itself or not.

This is very abstract (since boxes don't contain themselves) but when that condition is present the box would contain itself. It would not however be able to contain that condition as well.

If however the self-containing function was triggered by some condition and then operated to contain both the box and the condition the two would always be present together and inseparable within the system.

Whichever way this gets looked at self containing creates an odd man out.

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This is the ambiguity problem in symbol systems.

A symbol system has two types of symbol: the entities and the operators. For example

1+1=2

Now the system cannot operate on itself because + and = only operate on numbers, and the list of symbols above is an expression.

In computing for example we can express this expression as a string using a new operator the quote marks so

"1+1=2"

is an expression. Now quote marks operate on expressions so we can quote itself

""1+1=2""

However now there is ambiguity. Does this mean "" expression and 1+1=2 expression and "" expression written side by side or does it mean the string containing "1+1=2". It is not explicitly decidable.

This is not self-referential as only a copy is contained, and an inferior copy to lacking the outer most strings quotes.

To really contain itself it needs to contain a reference to itself, say:

S = "S"

But there is ambiguity again because both the S things are symbols. Is this saying that S is the string containing the symbol S (character 83cin ASCII) or the string containing the string called S. Is S a character or a reference to a string?

if we were to PRINT S in a programming language we would just get an output of S.

Another possible output is the recursive """"""..."""""" where S keeps being replace by "S"

But the real thing we wanted to express was self membership which can't be expressed in this language.

The ambiguity problem like the self-operation problem above requires an external switch to decide whether to treat the entity as a reference to the system self, or as the entity itself.

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In case self-reference seems void as it appears to be in these sterile examples. Consider a robot which like the ones in Japan can play football. When it picks its team it needs to be able to model itself as a member of the team, otherwise it will pick 11 players and try to play along side them making 12!

However it is not good enough just to model a team of 11 robots in its system... it needs to operate on the situation of just one of those robots (itself).

ok phone call...