SRH – Accurate definition at last

It is ridiculous how long it takes me to make even simple progress in these things.

The SRH states simply that in the event that something depends upon itself for its definition then there is the possibility of an infinite regression and an indeterminate result.

That is all I was trying to say all these years. So it is not that weak self reference of the type “This sentence has six words” is impossible. That sentence only describes itself, it does not try to define itself and so it is simply a false statement about itself. A false sentence is still a sentence with words so there are no definitions under threat here.

On the other hand a sentence like “This is the sentence with six words” has strong self reference because it defines itself as having 6 words when actually it has 7. In this case the sentence fails to be self-referential because it refers to something that doesn’t exist. In predicate logic there is no x such that x is this statement. It still intrigues me that a statement can be true in one form and then false when mapped into a different system of symbols. The Godel number for this sentence has different properties and so while in a 1:1 mapping with this sentence it has a different truth value if self-referential. Something up there.

However the most famous strongly self-referential statement “This sentence is false” one assumes by default is true but then is told is false. This completely overturns the definition. It is strongly self-referential and resolution enters an infinite regress.

A recursion example of strong self reference and why I’ve a problem with it.

int Me(int input) {

       return Me(input);

}

This algorithm definition depends upon itself. When called it runs forever because there is no non-self referential way of escaping the loop. Its value should be called NULL. What actually happens is that a block of memory is filled with the CPU instruction to go to a position in memory and also the data of its own position in memory. (I don’t understand enough at the moment to know how the chain of returns is set up so that if a value is ever found it will be propagated back to become the returned value of the function). It keeps copying the same block of memory into the CPU without immanent end … obviously the operating system can refuse to give the virtual program any more resources or if it is a direct bit of code the machine can be unplugged … so it is only an infinite loop within the context of the CPU’s game.

This fits into the Turin stopping problem. But can we say something like without self-reference a program will run out of space and have to end. Only with feedback can the possibility of the infinite be approached in a finite machine. Now not all feedback is self-referential and not all feed-back leads to an indefinite system… is there a relation here between strong self reference (SSR) and infinity and weak self-reference (WSR) and the finite.

Will need to understand a LOT more and run lots more experiments on the computer to get a feel of what in the hell this is all about. Will post some of the fractals I’ve been inspired to generate by the MIT course… last did this when I was a teenager!!