Set of All Sets and SRH

Sure must have been here before but aren't these deeply connected?

The "Set of All Sets" leads to contradiction via Russel's Paradox. As a result this entity cannot exist in most forms of logic.

Now SRH is saying that Russel's Paradox is the same problem that crops up in Godel's Theorem, Turing Halting Problem, Chaitin's Constant or just the simple Liar Paradox. That is the existence of "self reference" causes contradictions and breaks the system. Now its not that simple, as many forms of self reference are not problematic. The most recent speculation is that if self-reference is problematic then any theorem using self-reference is problematic. And SRH is therefore problematic as it is making a statement about the validity of self-reference itself. It is quite possible that SRH is undecidable. It is impossible to say for sure whether self-reference is going to cause a problem or not. All said intuitively, but not proven.

But let switch that and say "the Set of All Sets" is the problem, that then breaks all other problematic expressions of self-reference. Whenever we are trying to say "All" that encompasses itself we have this problem. In other words "I" cannot construct the "Universal" or vice versa. Or there cannot exist a particular member of the Universal that represents the Universal.

∀x¬∃y | x = y OR x ∈ y

I think that says "for All x there Does Not Exist a y such that x is y OR x is in the set of y"

If we say that two entities are the same if they are the same or have the same elements, then there is no entity that is either the same as everything or contains everything.

#TODO now show that all other problems like Liar Paradox are like this...

In all the below 'x' is within the set of countable numbers.

"I am false"

We could create a function g(x) taking a single number which represents the expression numbered by x. The set of all possible expressions is countably infinite, and so g(x) will map each number to an expression.

Now obviously in use expressions get bound so there are an infinite number of bound expressions for just g alone e.g. "g(9906753)". So we need make the Use/Mention distinction of Quine. Which is itself a form of SRH. The Use/Mention distinction occurs from SRH itself. If you mix them you get contradictions. Meta statements are exactly this distinction. Talking about something, must be different from that thing else you have contradiction. #TODO so in fact SRH is used even to express a self-reference!

Anyway, using g we can make this sentence:

∃x | ¬g(x)

There exists an x such that the sentence coded by x is false. No problem.

Now let g'(x) be the inverse of g(x) so that g'(g(x)) = x

Now suppose it were possible to make a meta statement like this: 

σ = g'(∃x | x=σ AND ¬g(x))

It says σ is the code for the sentence which says that there exists a number x which is the code for a sentence which is false and which is also σ

This is the liar paradox. But it is also extremely badly formed.

Try another approach. Suppose there was another expression I which is interchangeable with the number of the sentence that mentions it. So for example:

∃x | x = I

Assuming this sentence is valid then it must have a code number. That number is 'I'. So we are saying that there is a sentence which has the code number of this sentence. Like the Ontological Proof of God this must be true. And so we have a problem:

¬∃x | x = I

There must also be this sentence created by adding just a ¬ symbol which says there is not a sentence which has the code number of this sentence. Well there is, it is there. So we must reject such an expression as I.

No cannot be a component of a well formed sentence that refers to "this" sentence. No time here, but this is essentially the reason for Use/Mention. You can't go doing meta work outside a sentence like determining its code number, and then go using that within the sentence. It's odd to think that Von Neumann computer architecture has circuits for the 2 types of numbers Data/Instructions because of this. I suppose its a well known fact that if computers start writing into memory that contains code it's only a short matter of time before the computer blue screens or hangs! Now this is exactly what Turin machines do, but even they do not write into the machine state - that is still hermetic and sacred. 

Not all meta work is problematic. Consider this working sentence: "This sentence is made from 0b101110 characters". The difference here is that many sentences have 46 (0b101110) characters and many do not so negation does not force things "into a corner." But the g(x) meta function uniquely mapped to the sentence so ¬g(x) unique does not and so there is "no room to move." 

So anyway the formal Liar Paradox is off to a bad start.

#TODO you were originally looking to show that Liar Paradox can be reduced to problem of making claim of ALL