SRH – and Hermetic Self-Reference is impossible - progress?

Was thinking last night about Gödel Numbers.

I’m sure I’m making a naive error but it occurred to me that for a Gödel number to map uniquely onto a Statement they must both have the same information.

Whatever that means when it been made it rigorous doesn’t it imply that a statement can’t refer to itself directly since it cannot encode its own number without increasing its information content and entering an infinite loop?

This implies that in this understanding of Gödel a “reference” is established when a direct relationship is shown between two structures. They occup the same position in the parallel calculus and so have the same information content!

So it leads me to a better analysis of reference. What was termed Strong Reference and Weak Reference is better termed Hermetic and Non-Hermetic Self-Reference.

Hermetic Self-Reference (previously Strong Self-Reference) is what the SRH says can’t exist. Non-Hermetic SR is fine.

Hermetic self-reference exists when a structure fully maps onto itself without information being implied by the context.

A Godel Number would be Hermetically Self-Referential if Godel Numbering has a Godel Number itself that could be encoded within Godel Numbering. But as suggested above this is impossible. I still need to find the original paper – I studied it at college but fell short of understanding the approach directly … something about using ‘’’’’’’’’ (primes) to determine Godel Numbers. Can it really work?

This also opens up a new lead. The idea of information entering a reference from “outside” the system. This is what I’ve looked to show since the beginning.

So my reasoning began today at lunch.

In a 1D universe of 3 static objects the total information is easy to measure: n! = information of 6. I’ll take information as the total number of posible states (ignoring proper information theory).

However in this system of 3 things it cannot encode its own information because it can only ever have 1 state. To encode its own information at 6 requires the whole system again and so an infinite recursion is set up.

This is like creating a Quine that doesn’t output its own code by rather represents its own information content?

It’s a paradox. A system can only have 1 state. To say it holds “information” is to compare it with another system so that there are multiple possibilities.

So to say “I” meaningfully and encode information is always to put “I” in reference to other things – so it isn’t self-reference.

“This sentence…” acknowledges other sentences to make itself meaningful so its self-reference is not hermetic.

I’ve made this point before but it is getting more rigorous at last.