Quote:
Originally Posted by Buffy
Again, please state why this is a non-trivial assumption.
Stating that it results in infinite regress is not a valid conclusion, so I understand why you try to layer on this as an "additional implied assumption":
Although I entertained your assumption that the theorem incorporates this infinite regress, in fact it does not.
The input to H is simply a function and its initial input. There is no mechanism either implied or required that that "input" causes a recursion: that is, if you do use H as its own input, the parameters have no way of knowing that the input is code to be called rather than being data.
Again, realize that the theorem refers specifically to algorithms and not (necessarily) to machines running those algorithms.
Do not bite at the bait of pleasure till you know there is no hook beneath it, 
Buffy |
Your definition of trivial assumption needs revision badly. A trivial assumption is something that could not possibly be wrong. No proof is ever required to show that an assumption is not trivial, it works the other way around.
<Yawn> The supposed halting machine takes other Touring machines as input. Machines and algorithms are theoretically equivalent, so it doesn't matter which one you talk about. The algorithm can take another algorithm as input. The halting machine/algorithm recognizes it's input as another machine or algorithm. That is it's job. It takes encodings of machines or algorithms and determines if they are going to halt. To say that it can't recognize the encoding as a machine or algorithm directly contradicts it's definition.
Quote:
Originally Posted by Buffy
I appreciate that you are left to wonder this, however your inability to conceive of other approaches is not sufficient to make a substantiated claim that this is the only approach, which is exactly what you are doing!
Right, but you did use it as your claim for infinite regress. In order to substantiate your claim you must also prove that non-tracing approaches *also* result in infinite regress...
An undefined problem has an infinite number of solutions,
Buffy |
You seem to be forgetting that Touring started this conversation. The man comes up with an arbitrary manipulation of a supposed halting machine, and then claims that it creates a contradiction such that no such halting machine could occur.
Using this tactic an infinite amount of false "proofs" could be created. For example
1. My mother makes pies with alcohol on sunday's
2. It's illegal to sell alcohol on sundays where my mother lives.
Conclusion: My mother doesn't really make pies with alcohol on sundays since the local law really is against it.
Well, here we have assumed (but not recognized) that the mother couldn't have bought the ingrediants on a different day, a different place, or illegally, one of which is obviously the source of the contradiction. This is the form of proof that Touring has used.
Your claim is that I have to prove that the mother obtains pie ingrediants on different days or in a different place, or illegally . This is incorrect. The proof was faulty to begin with, so I don't need to prove anything.
But here you could say, it is possible to prove it, and since a bunch of non logically discriminating people believe it, perhaps you should. This type of thinking, besides just being wrong, creates a serious problem. It is easy to come up with such a proof where the assumption could not be proven one way or another. Example:
1. My mother died
2. My mother is in a better place.
3. Dying is a terrible thing to happen to someone.
Conclusion: Being in a better place is a terrible thing to happen to someone.
A contradiction has occured so it must be false that dying is a terrible thing to happen to someone or that my mother is in a better place right?
Wrong - I left out the assumption that people continue to exist after dying. This can neither be proven or disproven. It not only invalidates the proof, it makes the "proof" practically nonsense.
