The point is that you can often create a rule that defines a set such that the negation is very much harder to define in itself. Take the Mandelbrot set. The white area is easy to define: it is all the points which drive the algorithm to a magnitude greater than 2. We can calculate this. The black set is what is left over. Now I believe we can construct the Appleman directly from the fixed points of the iteration but its much harder. (Kind of playing with the fact that white points are computable cos they compute in finite time, while the black points basically never halt). The Mandelbrot set is thus by definition non computable looking at the black. However you can deduce it from the computable function of the white. Now is there a computable way to define the black?
I need think about this but I'm borrowing in part from Douglas Hofstadter who discusses this. Negating a set can create a new set which is hard to define.
Now for SRH the point is that you cannot create a set whose negation cannot be defined in itself, that is without reference to the original set!
A set whose negation is only definable in terms of the original definition is a problem. That is the hypothesis.
In the Jewish analogy it means you can construct Gentiles without reference to Judaism! That will comes as a challenge to Judaism because in Judaism the whole world can be built from belief in God and Jews. The idea that you can construct things from outside Judaism and God is catastrophic. But that is what I meant at the start (previous post) about being Hermetically sealed. It is essential for Jewish belief that there is no Outside. And it is that rejection of the outside which I believe is the cause of ALL the suffering that the Jews identify.
Just restate this side of SRH.
So if some formula f(x) defines a set F there must be a formula g(x) which defines ¬F where there is no h(x) such that g(x) = h(f(x))
That is to say there can be no theory that uniquely defined f(x) and ¬f(x). ¬f(x) must be defined by functions unrelated to f(x).
And this goes into that issue of what dependency really mean. In what way does a function contain another function or depend upon it?
We can illustrate with some maths.
Take f(x) which is defined x:= x^2 {x ∈ N}
F = {0,1,4,9,16,25,36,...}
¬F = {2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,...}
Now this hypothesis says that we must be able to derive ¬F without using f(x) that is a formula that does not use x^2 or can be reduced to a form with x^2 i.e. no x^4.
Now we know we can create a Turing machine that will take any input and create any output. So there must be a formula that will compute ¬F. But the question relevant to SRH is can we show whether that formula entails accessing the set F.
For example the algorithm I used to generate ¬F just counted from 0 and removed numbers in F that is I calculated the complement of F. But we're saying there must be at least a way of doing this without reference to F.
Wow that is quite a statement. I need think whether it can be proven or disproved.
Anyway if it turns out to be true, if we map it back into Judaism it will turn out that Judaism was actually the path to the Truth but ironically, and so J(x) remains central to the world!
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Lets see what this looks like:
Let f(x) be a set definition. It decides {-1,1} for all x.
Let g(x) is defined as -f(x). Simple. f(x) = -g(x) for all x.
Now we are saying that g(x) must have another definition h(x) = g(x) such that there is no i() where i(f(x)) = h(x)
Ok as written this makes no sense cos an example of i() is simply the negative function.
What we need is a way to express the idea that h(x) does not compute f(x) at any stage.
Now didn't we already show that such an algorithm is impossible. So it is impossible to tell if h(x) does or does not computer f(x). And therefore this hypothesis is meaningless!
Just check this:
So the simple proof was (after Turing):
Suppose there is a function C(f,g) which determines whether f is composed from g. That is if in calculating f(x) it involves computing g(x) at any stage.
Then we can construct the following function:
B(f,g) = C(f,g) ? 1 : g(1)
Now what does:
C(B,g) mean?
C determines whether the function B involves calculating the function g. But if it decides that B does involve calculating g() then it actually only produces 1 and if it decides it doesn't then it ends up calculating g(1).
There is a smarter proof involving currying to match the arity of the functions but this will do.
So having a function that can determine the composition of functions leads to a contradiction. This is the end of the road for SRH based on composition and dependency. You cannot determine in a formal way the composition or dependency of a function, and so you cannot determine if it involves itself. This is also the end of the road for the Universal Engineer or Eva. That was the idea of a program that could work out the the "faulty component." If you can't even universally work out from the output whether a function entails another function then how can you isolate the fault! The only way to do this kind of testing/debugging is to "open the program up" and isolate the output of its components. You can't do it from the "outside." This is almost reverse SRH. we set out to show that there is no intrinsic "self" hidden within a system, but did that from the perspective of the entity: any entity require an outside:
>that was Horatio "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy" meaning that all system are like Horatio's and cannot be total for they require an "outside" which is not accounted for by the system.
>+1 Hypothesis was similar that all systems will fail at totality because they will miss out themselves. So we can show that the "components "of a system are opaque to a viewer, it means that no system looking at itself can ever know its own construction!
AIME1987 (Artificial Intelligence Mechanism Emulation 1987 ) was a super naïve AI based around language structure. It wasn't a self learning machine, it was just programmed to asked syntactic questions about your text input so it could generate sentences itself. It never got to the point where it could ask itself questions. Primitive and useless. But it led to AIME1996 which was a pattern analyser that sought to reduce your input of a binary sequence to simpler geometric descriptions (kind of fourier). It was then fed its own state to see what would happen. This was the original insight for SRH. It started from particular speculations in 1990 that self-consciousness was linked to self-awareness and processing on self.
But all this appears at this stage to have just ended up showing that all systems have an inside that is not deducible from the outside.
Which is a trivial result.
Does f(x) = {1,2,3,4,5,6...} for x ∈ N arrive from returning just x or from returning x + 1 - 1
An algebra function can determine through formal steps that they are equivalent. But we are asking for actual function composition so we could determine "self" dependency where self was a particular application of a function where f(x) -> x+1 and g(x) -> x - 1 are examples of actual functions.
Regarding self its interesting that this echos our experience of "other people" that we don't know what is going on inside, and we see other people as a black box. That sense of other people being "black boxes" is very much the foundations of the ideas of "privacy", "self" and "personal identity." It great that no machine can ever deduce from the outside the true nature of our inner workings.
But it raises the question of what we know about our self. We don't view our self as a black box. But do we really see ourselves from the inside, or are we just viewers on the system from a different perspective?
If no system can universally deduce composition of another system, then it can't itself! Isn't that SRH! we are always on the outside!
Interesting that this current thread began with speculations on the Hermetically Sealed nature of the Jewish identity. I don't know if its is true, but Jews have a view that there is an inside and and outside to the world. There is the closed world of Jews and the outside world of Gentiles. Proofs on this page suggest we are all outside every construction, or at least we are unable to deduce the composition.
But is there still an "inside" beyond deduction. Can you still argue that "I AM" (Yahweh). You don't need to deduce yourself because you ARE yourself. But we speak of self knowledge, this must be impossible if self is "inside." If there is a self it is unknowable. Perhaps we can call that Yahweh? Altho this now sounds like Eastern Philosophy. There is no discernible difference between God and Man. This does not mean that God and Man are the same, but it means we can't know. We can't deduce composition from the outside we need take apart. The Original Sin suggests to Jews that they are separate from God. This means they do know.
#TODO Need to digest.
We also showed recently that a similar contradiction occurs in the more specialised function that looks for self reference. A function that determines whether the function is being passed itself is impossible.
#TODO Check this proof too.
So the foundations for a formal description of SRH are slipping away :O
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Okay what about a function embedded in another.
Take that function C(f,g) again which determines whether f is composed from g. That is if in calculating f(x) it involves computing g(x) at any stage.
Then we can construct the following function:
B(f,g) = C(f,g) ? 1 : g(1)
Now what of?
C(B,C)
Does B require the computation of C. well by definition it does. (is there another way to calc B without C?)
C(B,C) = 1
But we said that C() was not computable.
So although C entails a contradiction we can still define a single value of C that must exist from definition because of our self referential definition that it is used to build B.
This also means that in the case of self-reference we can know that we ourselves compose a function.
This is now Descartes. Given that we exist (or are presented by the ontology that is by definition) we can use this to construct some truths.
So a function can use itself as an axiom, or building block for logic.
So if we take any function f(x) then we can build f(f(x)) by definition.
Self-reference becomes the basis of an infinite inductive construction. But only one way
if g(x) = f(f(f(f(f(x))))
C(g,f) = 1 is fine through definition.
But given h(x) we can't work backwards as to whether it is constructed from f(x).
In fact there must always a function that is derived from itself, and so we can construct an infinite number of functions from self-reference alone.
So there is a sense of self which means just "the given entity."
In English we often start a sentence with "I am going to the shops". This is like defining the function me(x) where me can consume any number of things and we set x = "go to shops".
The sentence introduces the axiom of "me", and that then give the rest of the sentence something to build upon.
Where does this "me" come from? well it is like f(x) above. It is just a starting point, or basis for recursion.
Now we can go forwards and use me(x) perhaps there is the form me(what/verb, to/noun) so that the above is written "me(go,shops)". And it can get recursive and self-referential "me(wash,me)" or
if "me(write about, holiday)" then we can have "me(write about,me)" and if we can even write about our writing so "me(write about,me(write about,me))"
But we can go the other way and investigate "me(x)." It is a given, and it is a contradiction if we could know about "me."
So this is original SRH actually. The idea that there is a given that cannot be based upon itself.
To summaries the way here is to realise that you cannot work out the composition of a thing. You need open it up to deduce the real contents. It means we cannot work out what is inside the self. But we can go the other way and use components build other things. And our life is the process of using the self as an axion to build constructs. But it can't build itself.