"consistency" is the general mathematical principle. “Simple consistency” or “Negation consistency” is the form that you are talking about. It is not “true” consistency, it is simply one form of mathematical consistecy.
Omega-consistency is not a red herring. Omega-consistency is exactly the property that holds for sets that meet the standard for Godel’s original proof, which also happens to be the form of the Godel statement that you have been using.
And there is no place to move on to. You remain dedicated to your ignorance and dishonesty. I remain disgusted by your casual disdain for even the most basic elements of ethical discussion. And GIT remains unrelated to the virtual fantasy in which you imagine that you live.
The definition ultrafilter claimed defined “consistency”… doesn’t.
(Thank you for finally providing a citation, by the way.)
You seem to be ignoring the first half of those descriptors, Spiritus. A system can’t be said to be consistent just because it doesn’t contain a specific statement. As you’ve pointed out, it can then be said to have “absolute consistency” in regards to that statement only. The concept puts no restrictions on what else the system can be capable of showing… which is why it’s not generally used.
There’s a reason “simple” consistency is the one people spend so much time talking about – it’s the case where the system is completely consistent. That’s the strongest of the concepts you’ve referenced, and it was the first to be generated.
If you say that a system is consistent, standard-English rules indicate that you’re referring to “simple” consistency, otherwise known as “consistency”.
As the term generally refers to the property of an entire system, and “absolute consistency” refers to specific statements in that system… well, I would think you should be able to tell the difference.
An intellectual honest person would admit that a system is not consistent merely because it doesn’t include one or more statements. It may have absolute consistency in regards to those statements, but that’s about all we can say.
The importance of GIT lies in its demonstration that a sufficiently powerful consistent system cannot contain either a proof or a negation of all statements. Absolute consistency has very little to say about GIT. It’s such a weak property that there are few conclusions that can be drawn from it. It includes the sufficiently powerful systems, and many more systems whose power is much less. The only system it excludes is the one containing all statements… which has no power at all.
You’ve demonstrated two things:
You know much more about obscure mathematical terminiology than I do.
Your ability to use this terminiology to describe systems meaningfully and interpret natural-language statements regarding those systems in those terms is woefully inadequate.
Now that you’ve actually managed to produce definitions of these concepts, would you care to explain what they have to do with GIT?
And there is no place to move on to. You remain dedicated to your ignorance and dishonesty. I remain disgusted by your casual disdain for even the most basic elements of ethical discussion. And GIT remains unrelated to the virtual fantasy in which you imagine that you live.
Darth Nader, I’m sorry, but I don’t recall anyone forcing you to read this thread. The thread title is sufficiently neutral not to cause anyone distress.
Spiritus, you’ve only done the following:
Used the term ‘consistency’ incorrectly.
Used ‘consistency’ to refer to a variety of mathematical concepts that are much weaker than actual consistency (by leaving off the necessary parts of their names) thereby muddling up the debate.
Failed to make clear why these alternate definitions have any relevance to the ongoing debate.
Failed to actually refute the points made in the OP, instead merely repeating that “they’re wrong”.
Glossed over your errors, claimed they weren’t errors, and repeatedly highlighted ‘errors’ in my arguments that were either already dealt with or whose answers were obvious given the points previously established.
To paraphrase Arthur C. Clarke, you’ve been educated way beyond your intelligence. You use your ability to juggle mathematical terminology to mask your utter lack of comprehension of what those terms imply, and you hide your inability to rationally analyze natural-language statements beneath a veneer of accusations and pointless objections.
When was the last time you actually had a coherent objection to any of the original arguments?
You’ve spent the last handful of pages making that rather stupid argument that since all things that are consistent have property X, all things with property X are obvious consistent. (You then fell back on the claim that ‘consistency’ includes a wide variety of definitions… pathetic.)
While I can see no way to determine the precise nature of the universe, we know that the principles underlying it must be capable of generating the things we see.
Arithmetic is one of the things we can see.
Ergo, the world must be at least powerful enough to represent arithmetic. (The world might not actually be infinite, as is technically required, but it’s not difficult to show that it would be sufficiently powerful with an infinite memory. That can be done for good old vanilla computers (the handy devices, not whatever computes).
Is the world consistent? It certainly seems to be. I don’t believe anyone’s actually ever observed two mutually exclusive states to be simultaneously true (quantum mechanics included).
TVAA, give up. The sheer eloquence of Spiritus’ most recent posts has clearly decided this argument in his favor.
Well, that and this latest line of reasoning is utterly looney. You can’t make observations about the whole of the universe until you have observed the whole of the universe.
Well, you can, and you may well be right if the bits of the universe you are seeing are representative, but you are right by accident.
All I have to do is talk about the observable universe… observable in every sense, not just what we see when we use telescopes.
I don’t have to account for the things that don’t interact with anything we interact with, because those things don’t exist by definition.
erislover: What system describes the configuration of the universe and the interactions that allow “later” and “previous” configurations to be derived from them? That’s what the world is.
Are you waiting for me to show that the world is actually a giant computer game being run by evil robots? C’mon… you’re smarter than that.
The whole point of this discussion is that the behavior of an “abstract” system that reproduces the world is the world. The two concepts are really one: there’s no way to distinguish between them.
Those medieval religious fanatics were wrong because there’s no way that the memory of God is different from “actual” events.
The statement in the other thread about this being the mind of God and GIT placing limitations on the behavior on angels was the most right thing that’s been said in these threads so far.
robertliguori: Just by talking about “the universe”, I’m making a claim about the things I’m discussing.
It doesn’t matter if I talk about a galaxy or a bathtub eddy, they’re still things with certain inherent properties in common. (For instance, they have properties… all things that exist do. But there are things that don’t exist… or more properly, there aren’t things that don’t exist.)
Given enough computer power, I can run a simulation that duplicates those people to any arbitrary accuracy (I might run into problems with the Uncertainty Principle, but then the people themselves don’t contain any data below that level, do they? ).
Those people could then be placed in a simulated 747. No problem.
Give me a computer powerful enough, and I will simulate the world.
How many particles will that machine contain ? How do you intend to make it fit inside itself ? Will the simulation be realtime, or are you going to have to do some crappy old finite-step approximation ?
).