1) we may construct a formula grammatically by rules of formation (recursion), and
2) also by deduction from axioms using rules of inference.
This gives us two ways of constructing a sentence and this is the problem. We can see that a formula like:
1.0 ∀ x : x^2 >= 0 • x ∈
is a WFF in standard predicate logic and set notation (assuming I've got my grammar correct ;-) While:
2.0 x x : ^2 = x 0 ∀ > • ∈
is just nonsense in any notation that I know (tho one could create a grammar to which this conforms)
Now whether statement 1.0 could be arrived at by a sequence of transformations on axioms is another question.
BUT I answer my own question because consider:
3.0 ∀ x : x^2 >= 0 • x ∈
Statement 1.0 and 3.0 are both grammatically correct but while there is no integer which is an exception to statement 1.0 it is easy to find a complex number e.g. 3i whose square is -9 and so contradicts the statement which states that x*x >= 0 for all x. Statement 1.0 is true while 3.0 is false.
So being grammatically correct, provable and true are all different. Showing that provability and truth are separate is what Godel did. He created a true statement that was by definition not provable. It is fascinating, as I begin to take this in at last, that he has a statement which by virtue of what it says about itself fractures the rest of the system.It is reminiscent in my mind of the ontological argument for God's existence where it is argued that because God is perfect by definition, it implies that by definition he must also exist since not existing would be an imperfection. Can we really pop things (as Douglas Adams jokes in Hitchhikers) into and out of existence on the basis of logic and definition?
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just found this blog which suggests that godel's 1st incompleteness theorem is the SRH ... but I think its not that simple
http://www.cosmicfingerprints.com/blog/incompleteness/comment-page-2/
yes this is how not to prove the SRH... but first time I've seen an attempt to show it.
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http://www.earlham.edu/~peters/courses/logsys/g-proof.htm
YES!! In 30mins I understand Godel's theorem much better than before!! An excellent summary of the arguments.
Notice that we "prove" that G is true by metatheoretical reasoning, not within S. This is very important to many philosophers pondering the significance of Gödel's theorem and proof. It means that Gödel did something that the system S is demonstrably incapable of doing, namely, prove that G is true. Since formal systems are isomorphic with computers, and since the deficiency proved for S is proved for all systems above a certain threshold of power, one reading of the situation is that Gödel did something that no computer above a certain threshold of power could do. If so, there is something about human intelligence that can never be attained by any computer even in principle. How justified is this reading? Knowledgeable people differ sharply on this question.
G is the statement Godel discovered which states that it is not provable. If it is false then it is provable in which case we can prove a false statement (which is called inconsistent) - this is the bit that is likened to the "this is a lie" statement. If it is true then it cannot be proven whilst being true. Truth is thus distinguished from and lies outside the system of proofs.
My understanding at the moment is not that we can establish a fixed grand division between computers and Man, or Man and God, but rather that any system cannot exist in isolation - this is the SRH. It is not as cosmicfingerprints argues that there is a particular outside-circle outside all others called 'God' but rather the God hypothesis (previously suggested) is that any system (of sufficient complexity to refer to itself) is incomplete and implies a external system... ad infinitum as in Cantors infinity of infinities.
From its very inception the idea is inspired by disbelief in the idea that we can gain a single body of knowledge that is entirely comprehensive to the extent that it explains everything about even itself. That is to say that the Aristotelian Scientific project for example will not get closer and closer to a single dogma of truth that eventually can explain even itself, casting into the flames in a vast Humian Fork all that is absolutely false. Originally it was tried in 2007 as a refutation of materialism - in that the scientific process can't be explained in materialistic terms alone. But how to do this?
The method to approach the SRH has been to examine boundaries. A self-contained system has to incorporate its own boundary. Put another way: to speak of itself it must be able to identify and describe itself which means to place a boundary around itself, but a boundary that can be constructed within itself. I don't know if there are geometries for analogy that allow this - I've seen some very odd ones. Not got very far: only to analysis of circles and boxes within boxes at this time.
Self-reference (after which the SRH was badly named) raises the issue of boundaries as well. When we refer to something we are implying that something exists. To exist implies that it must have features which distinguish it from other things. To exist it must have some boundary that separates it from things that it is not, one must then find such a thing! If we cannot find such a thing, or argue for its existence, then we cannot refer to it truthfully. If we refer to ourselves the question then is how would be identify ourselves? or how would be separate ourselves from other things? or how would be put a boundary around ourselves? This is done by name in logic. But my unresolved query is that simply using a name doesn't really amount to full reference. I may use the name of Theodore Roosevelt (as I just did) but I wouldn't be able to recognise him and I know next to nothing about him. To what extent am I referencing "him" then? A computer can use that name as well (as it just has passing it over the internet) but does this text itself really reference the man? We assume that I am referencing it (as the writer) and that I am somehow acquainted with the man. This is a very complex avenue of enquiry one I have avoided!! However the question of the scope of self-reference and whether systems can ever comprehensively and totally self-reference remains. The SRH belief is that one needs a point of perspective outside a system to create the boundary - just as one needs to enter space to see the circle of the Earth, or Archimedes needs a place to stand off the Earth in order to move the Earth. Some leverage is needed in order to operate upon oneself it seems instinctively. Indeed that is a very satisfying analogy: maybe SRH should be Self-Leverage Hypothesis (stating that it is impossible).
Another analogy I read recently in a maths book. Euclid's geometry can be shown to be consistent if we accept that Arithmetic is consistent. In general consistency can be shown for any system but only in terms of another system that it must be assumed is consistent. There is no absolute proof of consistency: it is relative. This essential is the SRH - but it aint a proof.
The SRH is essentially the opposite of Godel. Instead of showing that a metalanguage (isomorphic with the object language) can make truth statements about the object language that the object language can't, the SRH seeks to prove within the object language the existence of meta-statements that it can't prove. Or something like that: metaphorically the finger on Earth pointing at the moon. Need time for this to all sink in.
to construct a sentence which refers to an entity which can only be constructed if we step outside the system.
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OK has some nice beer so bad time to be doing this but the pattern seems to be this any system which allows self-reference automatically allows the creation of a contradiction because it can refer to both itself and not-itself. Thus an infinite progression of ever expanding systems is yielded by analogy like the adding of G to the axioms, or each infinity implying a larger infinity... now how does Cantor do this? Is Diagonalisation the key trick?
Is there a general theory of systems that states that the better defined they become the more power they gain at defining things outside themselves... something like that this would be SRH... dream dream
The other thing is Tarski's undefinability theorem
http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language.
It seems that the semantics/syntax distinction captures the division necessary for the SRH just as the data/code distinction in computing which I first observed the SRH in. Lots of leads... will follow up 2morrow.
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