Spiritus Mundi

[TVAA]

I have to go to dinner… I’ll be back soon to continue refuting Spiritus.

[alice_in_wonderland]

*Originally posted by TVAA *
**I have to go to dinner… I’ll be back soon to continue refuting Spiritus. **

Oh is THAT what that’s supposed to be.

Right then.

[erislover]

*Originally posted by TVAA *
So you’re saying that this UTM is programmed to perform basic arithmetic, and it’s given a Godel number for basic PA?.. As far as I know, the UTM would perform basic operations on it… So what?

So the crucial point of GIT, the ambiguous Godel number, can be operated on (which we would expect, of course). Hence there must be something else you mean by there are states the TM can never reach because of GIT. What is that? Let me be clear:

1. What states? Once this is answered, then:
2. How do they relate to GITs?

For example: actualized TMs cannot prove things by induction except allowing for infinite time and memory. Since the universe as we know it is not infinite in time or space, we can probably safely conclude that even if the entire universe was a model for axiomatic mathematical systems there would be plenty of statements that can never be made, configurations that will never be entered, etc.

But what about GIT? That’s what I am still missing. You say, perhaps, that the TM can never generate a proof of a statement G. What does this say about not being able to accept all inputs? What does this say about crashing etc? Nothing, as far as I know. I fail to see the link.

What makes you think the universe is a sufficiently powerful system? That is also what I am missing. A lot of hand-waiving about equivalence without a decent presentation of it.

The UTM in question necessarily has more axioms defining its nature than mere PA.

Whatever, that’s not really in dispute, and as we are all aware adding more axioms to an already sufficiently powerful system doesn’t escape the ultimate conclusion.

Now, about the universe being that sufficiently powerful system…

TVAA:
Spiritus next point, in which he points out a common language error, is still incorrect. GIT does NOT show that certain statements have no proofs. It shows that certain statements have no proofs within specific axiomatic systems.

Spiritus:
Second, yes it applies only to “sufficiently powerful” systems. Was it really necessary for me to specify that since you have been explicitely arguing that the Universe meets exactly those criteria in order to justify your application of GIT?
Put down the etch-a-sketch. Take some ritalin. Stop pretending that each post in a conversation is uninformed by the preceeding context.

erl:
Next?

[Tuckerfan]

*Originally posted by jinwicked *
**Mmmm… you boys were at it all night, weren’t you?

And Tuckerfan, sure, but keep that damn girly poofy thing away from me. Hands only!**

What? You mean my tongue’s a “girly poofy thing” and I can’t use it? Oh well, hands it is!

[g8rguy]

*Originally posted by lucwarm *
**Gosh, if your team of experts read the threads and indicated that you were in the right, why not have them post here and explain (in precise terms) why you are right? **

Most probably because the mods around here frown on socks. That’s my guess; frankly, given the intellectual honesty TVAA has displayed in the last few weeks, my view is that the panel of experts is at best expert on unrelated disciplines (“my training in meat science tells me that Spiritus is an idiot.”) and at worst fictitious.

[g8rguy]

What? You mean my tongue’s a “girly poofy thing” and I can’t use it? Oh well, hands it is!

And may I add… damn, I need to find these geeky rants sooner!

[TVAA]

*Originally posted by erislover ***So the crucial point of GIT, the ambiguous Godel number, can be operated on (which we would expect, of course). Hence there must be something else you mean by there are states the TM can never reach because of GIT. What is that? Let me be clear:

1. What states? Once this is answered, then:
2. How do they relate to GITs?

**

1. GIT tells us that there are necessarily statements that the system cannot derive. Precisely what these statements are cannot be determined, although the classical example is a known case “This statement cannot be proven”.

There are almost certainly others, but I don’t know of any general methods for produing them, nor any means of showing that there necessarily are/aren’t more.

**For example: actualized TMs cannot prove things by induction except allowing for infinite time and memory. Since the universe as we know it is not infinite in time or space, we can probably safely conclude that even if the entire universe was a model for axiomatic mathematical systems there would be plenty of statements that can never be made, configurations that will never be entered, etc. **

Quite reasonable so far…

**But what about GIT? That’s what I am still missing. You say, perhaps, that the TM can never generate a proof of a statement G. What does this say about not being able to accept all inputs? What does this say about crashing etc? Nothing, as far as I know. I fail to see the link.

** First, I must re-emphasize that G is different for every system. It doesn’t refer to one, specific statement.

Secondly: The computer’s producing output is equivalent to it generating a proof for the outputted statement: it went through a series of pre-defined operations and generated a conclusion. GIT shows that there will always be statements that are true within the system that cannot be proven. What this means is that the computer’s output can never take certain forms. If the computer outputed G, that would mean that it could generate it from the axioms of the system – in order to produce G, it can only apply its operations to its input. We’ve already established that the axiomatic system of the computer cannot prove G, and generating G is equivalent to proving it.

**What makes you think the universe is a sufficiently powerful system? That is also what I am missing. A lot of hand-waiving about equivalence without a decent presentation of it.Whatever, that’s not really in dispute, and as we are all aware adding more axioms to an already sufficiently powerful system doesn’t escape the ultimate conclusion.

** This is the objection to my argument that I have the most trouble understanding.

The universe is necessarily a sufficiently powerful system because time exists. To put in another way: one configuration of the universe leads to another. That’s causality.

There are plenty of obvious sytems that can be used to generate arithmetic: the human brain, the movement of molecules, the behavior of electrons, sand at the beach…

We know that we can construct systems that generate arithmetic. Therefore, we know necessarily that the level of reality “below” what we build is at least as powerful at that.

Does that help any? I’m not sure if the basic point is clearly explained…

**Now, about the universe being that sufficiently powerful system… **

Isn’t it obvious that it is? :dubious:

[TVAA]

alice_in_wonderland and g8rguy: Don’t bother.

I’ve recently clashed with g8rguy, so I understand his hostility, but I know of no reason for AiW to be annoyed.

You are free to disagree with my position if you feel I haven’t sufficiently proven my point, but I will not allow you to simply ignore the errors I’ve shown in SM’s statements. Have you even looked at the links I’ve provided? The definitions and examples clearly contradict claims he’s been making about what GIT is and what it implies.

[TVAA]

*Originally posted by Spiritus Mundi ***First, and most obvious, you are not quoting Godel’s Second Incompleteness Theorem. You are quoting somebody’s common language interpretation of GIT[sup]2[/sup]

** There’s really no difference. You may find the formal version easier to understand, but informal statements can be accurately made about GIT. Again, if you’re not capable of using words to signify meaning, please just say so.

**Second, yes it applies only to “sufficiently powerful” systems. Was it really necessary for me to specify that since you have been explicitely arguing that the Universe meets exactly those criteria in order to justify your application of GIT?

** It’s very important. Subsystems of the universe are not necessarily sufficiently powerful; leaving out one of the vital requirements of GIT does not strike me as a good way to debate about it.

Put down the etch-a-sketch. Take some ritalin. Stop pretending that each post in a conversation is uninformed by the preceeding context.

One of the problems in this “debate” is that you’ve consistently ignored points made within its context. You keep demanding explanations that have already been given, and you make inaccurate and incomplete statements about what you mean.

[**Third, if you are going to insist that GIT must be spoken about in common language terms, then you lose the right to call people “stupid” for not listing all of the restrictions upon an application of GIT every time the subject comes up, you raging hypocrite.

** It seems to me that I’ve been citing those restrictions and requirements over and over again. I fail to see the problem. If people wished, they can always return to the original thread, yes?

**Fourth, despite the rage with which you quibble, you have not actually offered any rebutal to the conclusion. Either the Universe is not sufficiently powerful for GIT & GIT[sup]2[/sup] to hold, or you cannot ever demonstrate that the Universe is consistent. Do you think that me names somehow erases evrybody’s memory of the point under discussion?

** For some remarkable reason, you’re ignoring the flaws, errors, and inaccuracies I keep pointing out in your statements and focusing on the aspects of my statements you don’t understand. It’s a reasonable strategy, and one I would expect from a thinker of your caliber.

If nothing else, these threads have been eye-openers for me. I actually respected you before these discussions began.

The question at hand is whether the system of the universe fits the criteria for GIT. I’ve tried to explain why it’s necessarily the case that it does. If you don’t understand, ask for clarification.

That is stupid.

YOU are stupid… and despite what you may think, I find that the presence of this signficant psychological blind spot within your mind to be a tragedy. It saddens me to see an otherwise fine mind brought so low.

[butter_pie]

This was going so well…

what happened to the Turing machines?

[lucwarm]

You can quibble all you like, but eris spelled out the essential point here (IMHO):

You say, perhaps, that the TM can never generate a proof of a statement G. What does this say about not being able to accept all inputs? What does this say about crashing etc? Nothing, as far as I know. I fail to see the link.

[TVAA]

Isn’t it obvious? The system can always be given a set of input that corresponds with a demand to find a proof of a G statement, which the system can’t do.

[g8rguy]

*Originally posted by jinwicked *
**This was going so well…

what happened to the Turing machines? **

They got eaten by the invisible panel of experts from outer space?

Either that or they read this thread and disappeared in a puff of logic.

[TVAA]

Here’s your well-deserved reward, g8rguy.

[discontinuity]

[lucwarm]

*Originally posted by TVAA *
**Isn’t it obvious? The system can always be given a set of input that corresponds with a demand to find a proof of a G statement, which the system can’t do. **

Not being able to do something isn’t the same thing as crashing (or is it?)

In any event, the argument you make doesn’t require GIT. What if the system is given a set of input that corresponds with a demand to draw a square circle? A demand to find a real number x such that x^2 + 1 = 0?

Maybe the computer will start smoking and stuff just like NOMAD did in Star Trek? DOES NOT COMPUTE DOES NOT COMPUTE MUST . . . SELF-DESTRUCT

[FranticMad]

But a function requires an argument.
If the system demands a real argument, but the system cannot model real arguments, then a machine can generate new axioms until it simulates real arguments. Therefore, inputs with real arguments can cause an infinite loop which generates unprovable outputs.

Hence GIT applies.

[TVAA]

Have any of you ever read Godel, Escher, Bach?

[erislover]

yes

[TVAA]

Remember the record player that could play all records?

[erislover]

Yes I do.