So I've been insisting with someone they should keep a daily record of their symptoms (not coroanvirus). And we joked that the stress of keeping that record might set off the symptoms.
Now you could add an extra factor to your analysis, and extra row in the spreadsheet to record whether you were keeping a spreadsheet. However due to SRH this is meaningless. It would need to be always ticked. In logic it is a tautology and what is necessarily true is also meaningless. We might do some statistics using that factor but obviously we would never learn anything.
For this situation to make sense we need an independent recorder to make a new spreadsheet that records our symptoms and whether we are updating our spreadsheet. This row can now contain proper data with a decision being made on whether yes/no. That actual making of the decision is the basis of the data and meaning.
Now the patient can deduce all this from SRH because they know automatically that rows on their spreadsheet that are self-referential are open to issues.
But here is the crux. Not all self-referential rows are a problem! And this has been the sticking point for SRH. Can we determine absolutely a rule for knowing when self-reference will be a problem. The SRH was the hypothesis that there is a definite way to decide whether self-reference would result in paradox.
The classic example of self-reference not being a problem is in infinite sets. The Cardinality of Natural Number is a Natural Number. The interval [1,2] of the Real Numbers maps to the interval [0,3] thus a subset maps to itself. This result would appear to destroy the SRH. The Natural Numbers can be constructed from a natural number 1 and the S() successor function. It is therefore defined in terms of itself. Especially in maths there are examples of SRH that don't result in paradox.
The key concept I wanted to introduce here was "dependence." If we allow that a things have a logical dependence then its easy to express SRH. The classic sample from this blogs Past is "castles in the air" (CITA). The problem is that CITA are usually considered to rest on clouds. They cannot support themselves, and so are dependent upon some "ground." In gravitational fields things are "dependent" upon other things. Now its interesting to observe that a single entity while creating a gravitation field is not influenced by its own gravitational field. It seems that self-reference is indeed protected here. Yet that entity is really just a collection of smaller entities that are both creating gravity and being influenced by each other. So you can never really have a single entity here. Its also interesting to note that a pair of entities while not being influencing by their own gravity directly can influence each other so that they do end up experiencing the influence of themselves second hand. Why is this? explore later. That "dependency" creates rules that underpin SRH. Once something is dependent on something else, paradox arises if that dependency is ignored. The King standing in our CITA will fall to "ground" when the CITA does since he depends upon it for support.
Now let me get to the point. Assuming "dependency", the SRH says that there is a definite way to determine whether self-reference will be a problem and that is by looking at the "dependency." Dependent Self-Reference is the issue here. Now SRH itself must conform, so it would be able to tell in a definite way whether the Theory has problematic self-reference.
Now its like the spreadsheet above.
The theory depends upon itself in this scenario, and so it must always be without problematic self-reference and so tautological. Otherwise it is problematic and we don't even get to updating the spreadsheet. There is a more rigorous argument here but its essentially Halting Problem (HP) revamped.
In fact using HP suppose there was a decision algorithm that took a theory analysed and decided if it was self-referential in a problematic way. e.g. "the set of all sets that don't include themselves."
Incidentally while Zermelo Fraenkel amongst others proposed a way around Russel's Paradox what interest me (and SRH) is why the paradox is there at all. There are definite conditions that lead to problems, and as Godel and others afterwards have shown the existence of these always self-reference derived problems can be used to cause all kinds of problems. And you can tweak things to avoid the paradoxes, yet they're always waiting to strike. Godel actually uses SRH to provide his proof. SRH is much bigger than incidences of self-contradiction occurring its about the existence of such a thing at all.
p.s. the real classic was machines understanding themselves. This is linked to AI and self-consciousness. All very Hofstaedter.
So SRH takes a theory and spits out whether it is clean (result 0) or self-reference problematic SRP (result 1).
So following Turing we build a new machine (SRHb) that uses SRH to decide the input.
==NOTE SRHb is nonsense it proves nothing (logic below is faulty) BUT the gate is open. Should be able to complete this now.
But its much bigger than Turing because these machines are actually examining the input for self-reference issues. They would detect the presence of contradictions and report on that, rather than just cause halting jams. If you put Turing's Halting Algorithm into SRH it would spit out that it causes a contradiction.
Now you put SRH into itself what do you get? Does knowing whether things cause contradictions (by specifically by examining their self-reference SRH) itself cause contradictions or doesn't it?
The other implication of SRH is that simply by looking at the self-reference relations of an entity you can decide whether a contradiction is possible. The set of self-reference relations includes all the contradictions. There are no contradictions that don't depend upon self-reference! That is an essential corollary of SRH.
SRHb is then given itself. If SRH spits out 0 then it is not problematic, yet SRHb spits out 1 so it is problematic.
But SRH should be able to see this Turing Contradiction coming so it must spit out 1 for itself. SRH must then be self-reference problematic (SRP) which means that we must reject it. Yet if we reject it we are conforming to SRH. So SRH gets us truly stuck. We can neither accept nor reject it!!!
So there must be contradictions that do not depend on self-reference, so we can reject SRH on a bigger scale than SRH itself (since we are rejecting that self-reference is the problem). OR we cannot decide before hand by any means whether something will result in a contradiction. SRH is dead.
==== Update 25/9/2020
So there is a very obvious contradiction to note first.
Strong SRH (SSRH) says that self-reference always leads to contradiction. So what does SSRH say about itself? It says that it is contradictory to even ask! But this is worse.
Let us adopt function syntax to indicate applying a logic SSRH(x). So SSRH(x) says that if theory x contains self-reference it was derive a contradiction. So at first the above says that SSRH(SSRH) will derive a contradiction. But the very act of applying SSRH to itself is the cause of the contradiction. So if SSRH(SSRH) indicates a contradiction that SSRH is true. It is contradictory to even have a contradiction! There must be a meta-state that is beyond true/false (analogous to Turing Non-Halting) that SSRH(SSRH) implies. Interestingly this is the +1/Horatio parts of SRH. Self-reference has broken the system and implied a new "terms" or "domain". SSRH even breaks itself! I need think that through some more to tidy up.
There is also the weaker SRH (WSRH) to consider. The special conditions for self-reference I was suggestion before are the same as for Fixed-Point. Any system that has a fixed point, and which is self-referential will have a paradox. So more subtle SRH includes this Fixed-Point axiom.
