So, Oh God! proved that God exists?
It did about a good a job as you did, and was a lot more entertaining.
I give up, Spiritus. You win. The Gods themselves could not contend against you.
I did not win. You remain as ignorant as when you opened this thread.
Just in case you do reappear, you should know that, if anything, these unobtainable “memory sequences” correspond to the non-theorems of an arithmetical system, and not to the true but unprovable type Godel statements.
You must have played with a Rubiks Cube? There are some configurations that are just not reachable, say a single corner cubie twisted on it’s own*, does this make the cube subject to GIT?
Much of your argument appears to hinge on “unreachable states” <=> GIT, and that is simply not true.
*Actually you can get there by “stepping outside the system” and (for want of a better word) physically twisting it, but not by valid twists of the cube’s faces.
There are other configurations that just aren’t reachable.
For example, there can’t be a memory sequence that corresponds to the proof of the statement “this system cannot say this statement is true”, because if the statement were false, there would necessarily be a way of proving it (and so the system isn’t consistent), and if it were true, it’s unprovable.
And yet, every possible memory configuration in a register machine can be set and tested during diagnostic runs. Nothing stops the machine from reaching any physical memory configuration. This is trivially obvious. Every bit in a register machine is addressable. Every bit may be set “on” or “off”.
What does that mean? It means that not every set of configurations through which a computer can cycle can necessarily be placed in correspondence to a logical system of sufficient power to invoke GIT. In other words, TVAA’s basic premise is wrong.
It also means that being able to express every possible finite sentence in a system still doesn’t allow you to prove things that cannot be proved within that system. But I think even TVAA might have known that already.
TVAA – I guess to continue this conversation in this thread’s bastard offspring in the pit would be to post off-topic, so I’ve brought it over here.
I didn’t really understand your post in its entirety, but I notice that you latch onto the notion of consistency, again.
From page one:
(The parenthetical “more than” was unintentionally missed from my first post, oops.)
…That is to say, specifically, that there are no a priori reasons (that I know of – please illuminate me if I am wrong) to insist that any AI machine will have to instantiate a formal, consistent arithmetic.
I mention this because it seems your OP addresses AI directly:
1. Godel’s Incompleteness Theorem does not imply that humans will always be smarter than computers
I agree, as I suspect just about every other poster to the thread does.
2. It does mean that no finite system can ever derive all truths.
I believe it is true that no finite system could, but this is more readily seen as a consequence of its finiteness and not GIT.
3. No computer can be made that cannot be crashed.
This seems to me to be trivially false, and several examples of non-crashable computers have been given. Extending the definition of “crashable” to include “smashable by a sledgehammer” is asking too much of a reasonable definition, but even accepting such a broad definition, it does not then follow from GIT but more from the notion that everything can be broken.
4. Mathematics is necessarily incomplete: there will always be “true” statements that cannot be proven within a given system.
Indeed, this is a consequence of GIT, the truth of which nobody has denied.
But I’m confused now, because if 3. were true then wouldn’t that make 1. false? Or do you know of methods to crash the human mind too?
Point 3: The “computer” in question that would be smashed is a program being run by physical reality. In this program, the sledgehammer strike is merely input that the computer system can’t handle.
There are more interactions possible in physical reality than the computer can cope with.
There are indeed methods for crashing human brains. I presume you heard about the whole Pokemon-seizure problem?
Fair enough (but that’s not what is usually understood by the word crash), and precisely what’s that got to do with GIT?
TVAA, I really, really, believe that you have may some point, but I’m not following it.
To the best of my understanding the universe does follow from one state to another according to deterministic rules (except the possibly minor hiccup of quantum-mechanical dice-throwing (but even then there are still “rules”)).
Are you saying therefore that there are states the universe will/can never reach? I think that’s agreed – the universe will never be populated by nothing but overripe bananas, for instance – but again, what’s that got to do with GIT?
This has got to be a whacky aside – if the universe is wholly deterministic then the past, the present, the future are really just illusions, right? There are no degrees of freedom, just a single unified, unmoving snap-shot of the whole of space-time?
i.e. Exactly ONE state – all other states are trivially unreachable.
Ah, for goodness sake just tell me what you mean, I’m clutching at straws here.
** Sending a program into an infinite loop, causing it to use up its available memory and catastrophically fail, or otherwise forcing a computational system into a configuration its design cannot accomodate can reasonably be called “crashing” the system.
Well, thank you for at least giving me the benefit of the doubt.
That does indeed seem to be the case; while there are those who would argue the question, their objections don’t seem particularly meaningful IMO.
** Well, that’s trivially true if we consider the entirety of everything – it will always do the only thing it can do.
But that’s not the point. The whole point of this argument is it’s not possible to make a system that can cope with any data it’s given because it’s not possible for such a system to exist. This has a variety of consequences, including the one that any pattern within the universe will always have vulnerabilities.
For example, it’s not possible to make an immune system that always ignores the body it’s in but can react to every possible external substance. Very effective and accurate immune systems might well be possible, but there will always be a (hypothetical) interaction with the world that would cause the immune system to fail somehow, either by not reacting to a threat or overreacting to a harmless thing.
Of course, if the universe if finite, then there might be systems for which the circumstances that would cause them to fail never actually arise within it. In that case, the system would indeed be “perfect” – but no one would ever be able to show that it was in fact perfect.
If the universe is infinite, the possibility that any system will be confronted with its vulnerabilities is necessarily non-zero.
Is that any clearer?
So where does the universe get its data?
It doesn’t.
So what is the point of comparing the universe to a UTM?
I’m not, exactly. I’m showing that, at least for the case of UTMs, there is no true difference between the program and the system running it. SM’s general statement that the two are necessarily different is thus falsified.
How in the hell was that falsified?
SM says that there’s a fundamental difference between programs and computers.
I show that, at least in one case, that’s clearly false.
Since there’s an exception, the statement isn’t true.
I missed the exposition, I’m sorry. If you mean this:
Then color me unimpressed.
As ever, emulation and simulation are not indicators of the equivalence of existential identity. If they are, I don’t feel it is self-evident and I would request development of the point.
[sigh] Fair enough.
I’ll be back in a while to continue this point.
It might be worth a seperate thread. Just MHO.