Here's the rule: when something is applied to itself it creates meta properties that lie outside that thing.

Which is obvious. To apply "to" oneself is to already to set up an "outside" to oneself. Already blogged on that regarding the "dative" case. This is really going around in circles (irony given we are talking about non-halting, recursive self-reference type things). Need to consolidate all this at some stage.

Couple of conclusions:

(1) Can you refer to self, "within" self. That is get a handle "on" yourself from inside. Suppose you could (and I keep using proof by contradiction, while now aware that T/F are part of the problem of dichotomies that we suspect SRH causes). But anyway... suppose we could get a handle on self from within: a true Monad.

Well the first thing to note is that a self which has a handle to itself from "within" is also the universal set, in that if it had handles to anything "outside" itself then it would belong to a greater set and reference to that would be reference itself from outside. Universal sets containing themselves is already problematic. QED (informally).

(2) Suppose there was a formal way to decide whether a definition was impredicative. That is a formal way to decide whether a definition included itself. Given a definition d we can simply run through the procedure I(x) to discover whether d is required prior to definition.

Frank Ramsey gives the example of a normal impredicative definition "tallest person in the room." in that the selected individual T is present in the definition. However this doesn't work because in the definition T is selected as being in the room, and the definition then selected this person based on height.

The problem only occurs if the room was filled with people meeting the description "the tallest person in the room!" Given a group of people in room A you can't then put people in room B who meet the description "tallest person in room B." If you could you have the option to make a room C filled with the people who are not the tallest in room C, or lets go crazy and put everyone in room C who is not in room C.

But you have a good example there of how SRH can mess things up and the kind of problems that fall under SRH and those that don't. Level 1 is that if we allow SRH then we can create contradictions. Level 2 is that SRH allows or contradiction itself!

Anyway I(d) where d = "occupants of Room C who are not in room C" so I(d) is true. Note the tautology d' = "occupants of Room C who are in room C" is also I(d') = True. The point is that you need to know the outcome of d to decide on d.

Now let us decide whether our Impredicativity definition is Impedicative: I(I). This is fine, it doesn't matter whether it is T or F.

Now a new definition J(x) = !I(x) just returns Not(I(x))

I(J) = False

Initially we think that J is defined only in terms of I() so I(J) is False.

But J(x) is defined as Not this result. So J(J) 

I(d,e) -> I(I,I)

Jd(x) = !Id(x)

I(Id,Jd)

 TBC...

Note interestingly unlike the Halting disproof the very use of J sets the result, while in Halting the decision is "lower" down. There are "layers" of activity within the logic, its not just simple T/F.

====

Original Post...

So in functions that apply to their domain you get "fixed points." Now I dismissed fixed points in a recent post because they are actually elements of the domain that are unaffected by the function. They effectively "pass through" and so appear useless.

But what I have said before and missed recently is that this is exactly WHY they are important. The function does not know they are fixed points. Being a "fixed point" can only be determined from outside the function! No function/system whatever can ever codify its own fixed points by definition because the fixed point is unaffected by the function!

So fixed point by definition define the limits of systems. Where there is a fixed point the system cannot be total!

So you need to prove that there was a function that applied to its domain that did not have a fixed point and that is impossible.

So I retract the previous post. Fixed points are actually the point.

SRH thus works because when you introduce self-reference even so little as to apply a function to "its" domain then you create meta properties that lie outside the system.

Now I need to check Godel's proof again because IU have always claimed that Godel numbering lies outside PM. Is this true? If it doesn't then does it number itself?

Not all the impredicativity here. Lots of "its." When a function takes its own output as input then it is defined impredicativitily. Altho not endless recursion because it is defined by a particular element at time=0. But like rainbow tables (or the 100 prisoners) this leads to finite loops (usually except in chaos in the real numbers where the loop is countably infinite). 

===

Just to seal that point. Suppose there was a function f(x) with a fixed point of 5 that is f(5) = 5. Now we augment that function so that it takes it start value and compared with the outvalue to identify its own fixed points. But now its not the same function it is g(x). This new function outputs 0 or 1 for input values that are fixed points of the original function that is g(5) = 1. So now we have a function with a massively reduced domain. g(0) = 0 and g(1) = 1 offer the only two possible fixed points. And we don't know whether f(0) = 0 (which would contradiction g(0) = 1) or f(2) = 2.

So perhaps we have a more powerful function f(a,b) = {0,1} which outputs 1 if a(b) = b and 0 otherwise. With a bit of a cheat called currying we create a new function fa for each f(a,x) = fa(x). Enabling a 2 arity function to explore itself by splitting it into an infinite family of 1 arity functions.

f(fa,x) = 1 when fa(x) = 1 and when fa = a then this function is saying that 1 is a fixed point of f(a,b) 

But now SRH turns up cos we can create the paradox f(fa,0) = 0. That is there is an 'a' such that f(a,0) = f(fa,0) = fa(0) = 0 which says that 0 is not a fixed point of f(a,0) and yet it is. Therefore there cannot be a function f(a,b) which works out if a(b) = b reliably! Quite a weird result as it looks easy!!

But in the context of this post it illustrates how there is a fundamental divide between the meta properties and the thing itself. To violate this in general creates contradictions. Which is all very Constructivist and Predicativist.

Now again hot on heals of SRH. Suppose we have a theory T which proved SRH. Which is to say that we had a theory which proved why meta qualities are "meta" and cannot be encoded within the system. For example Quine's Use/Mention distinction : that is SRH is the theory which explain how Use and Mention are fundamentally separate. This theory would prove that it is has a meta level that is beyond itself. That is we can say things about SRH theory that it cannot express. And yet SRH theory is at the same time explaining how and why such a barrier exists.

 

In the Use/Mention analogy SRH theory would explain why Using SRH and Mentioning SRH are different. Yet to do this it would need to being Use/Mention together thus contradicting itself.

But what if it assumed that Use and Mention could be combined in am SRH theory which was then shown to be contradictory. We then reject SRH which proves SRH that they are separate.

OK rambling now but this perhaps looks not to complex now.

Just check the "exceptions" idea. So what if the SRH said that there are some meta qualities which can be encoded in the System (making them no longer meta) and SRH was that meta quality: it would be possible to encode SRH within itself precisely because it was a contradiction!

Doesn't SRH expand Godel statements to a total collapse of logic. The very idea of contradiction sanctions the contradictory theory called SRH which allows for a contradiction! No longer is Russel, Tarsky, Godel, Turing, Berry, Caitin and many others finding individual statements that cause contradictions, but the very idea of contradiction via SRH itself causes a contradiction.

It turns out then that the distinction of system/meta lies at the very heart of logic and is the origin of contradiction itself. Where we normally call A & -A a contradiction and reject at least one axiom to restore consistency, in a system that allows for SRH (self-reference in Godel) we can exploit meta qualities of that system (which cannot lie "in" the system by definition of being "meta") to cause contradictions, but contradictions are in fact only possible because we have meta statements. Calling something a "contradiction" IS a meta statement!

Take a function f(a) = 1 if 'a' codes for a contradiction, otherwise 0.

f(b) = 1 when b codes for this statement. This is nothing more sophisticated than the liar paradox "I am false." It says "I am a contradiction".

But we are only able to make a statement "about" another statement because of the "meta" level. "Meta" may as well be "about" or "mention". Without the possibility then we can't do this. This "meta" distinction lies at the heart of contradiction.

Ok am I getting beyond myself. "A & -A" is not making reference to itself via a meta level. It is simply taking anything A and then negating it and saying both are true. By definition this is always false.

But of course A can be anything even "A & -A" so it does have self reference! By substitution it evaluates to:

(A & -A) & -(A & -A) -> (A & -A) & -(A & -A) = (A & -A) & (-A | A) which is in words: a contradiction AND a tautology. Which is also False obviously.

But now we have infinite recursion. We can substitute it again and again infinitely and still get the same FALSE result.

= (-A & (A & -A)) | (A & (A & -A))

There are an uncountable number of ways to recursing and reorganising this FALSE statement. Not quite sure where I am going with this just exploring logical recursion deriving from the self-reference always present.

And this is pointless anyway as Tarksi already proved that TRUE/FALSE is a necessary meta level to logic.

But while he showed that allowing TRUE/FALSE To exist in the logic leads to contradiction, SRH is saying that contradiction itself comes from this distinction!!!!

Ok need think about this some more.

===

So its all about mirrors. To get a "self" you need a mirror. And that mirror image by definition is not self, as its separate from the original object. This gives us Use/Mention. The Use is the object, the mention is the reflection.

But given that a reflection is just a "copy" of the original how can you end up in Paradox? And I have read things which argue that paradox is actually impossible based on this kind of understanding. Well the problem is that the reflections are infinite. Its not just A and A', but (A and A') and (A and A')' ad infinitum.

"this sentence is a mention"

Subtle meaning. The "outer" sentence is in quotes so is a mention, but when read the "inner" sentence is in use. So while being read it is false, but as text on a page it is true. But a mention cannot have a T/F value until it is in use, so actually this is false.

"this sentence is a use" is true while being read inside quotes, but within the context of this whole sentence it has no T/F value and is just a quote. So the sentence is actually true while you assign a T/F value and then null when in quotes. Slightly confusing. Stuff in quotes is just "asfasdfasd" can be anything doesn't need a meaning.

So the SRH issue is just why these two levels. Why always 2 levels. Why True/False? Why Use/Mention? Why Description/Meta-description?

If SRH was to transcend these 2 levels, to unify opposites, to ignore impredicativity completely, to become the reason why we must have T/F distinction, it would need to be a theory that breaks itself, that sets up a reflection process that then reflects itself in itself and creates 2 from 1. SRH would be its own reason to not be itself. Contradiction is ONLY then possible. So we can ignore any logical contradiction proofs for SRH. It must be a language theory that ignore T/F to begin with.

Let us start. There is a theory Q. Q is an operator such that Q(Q) = Z. Z is fundamentally different from Q. If Z was the "similar" to Q then it could be shown that Q(Q) != Z.

So we are saying that Q is so constructed that it derives in someway a new theory Z with a "difference" from Q. Despite being derived from Q it is "different". This would be analogous to the NAND operator from which all other logic can be derived. But NAND is not itself a logic, it is a Truth Table which defines logic. A NAND B cannot operate on itself. Anyway rambling, just looking to see if anything comes out.

    

===