With all the bickering in here I’ve lost count of the votes for TVAA, you inconsiderate bastards. I have to start all over again. I’ll be up all night now, not that you guys give a damn about my problems.
Anyway, on with the re-count! So, here goes:
Uh, Zero…
Phew! Math is hard!!! I’m not good at arithmetic, as you can see. What comes after zero, should the occasion arise?
:rolleyes:
Some sources deal with some elements better than others. This is the nature of sources. No doubt TVAA finds sources that agree with his intuition and simply assumes that they must be right about everything. Then again, TVAA epitomizes sloppy thinking in many respects.
:rolleyes:
:rolleyes:
Of course it is useful. It makes proving many things much easier, that’s why “Idiot’s guides” tend to use it. What it also does is add another restriction (or set of restrictions, depending upon how “truth” is inserted into the argument) to the necessary qualities of systems to which GIT applies. This makes GIT LESS GOOD than teh formal result of Godel-Rosser, which is derived entirely as a syntactical consequence. This means that GIT can apply very generally to very many systems with very specific and well-defined qualities. Most mathemeticians and logicians consider that to be a very GOOD thing.
TVAA, being uninterested in such things but caring passionately that his pet preconceptions of reality be “true”, disagrees. But TVAA lacks both the understanding and the honesty to say that he wants to talk about “TIT”, the less powerful cousin of GIT that applies less generally but allows him to talk about “truth”. Of course, TVAA also resists taking a stand on what particular theory of truth he wants to assert. He considers that unimportant, apparently. He just really really wants to say that some Godel statements are true. (He seems to go back and forth on whether this is a general requirement or one that can be specifically applied to his favorite form of G, but if it does not apply to the G used in teh proof then it really really makes no sense to insert it into the system.)
:rolleyes:
Sure, except for that whole “diffeent level of reality” thing, right? And the spation-temporal locations.
And the component particles.
And the ongoing existential identity.
But, hey, as long as you are just using words in their “common dictionary definitions”. :wally
No rolling eyes here. On this point we are in total agreement.
:rolleyes:
You are an idiot.
I have told you the same thing numerous times. Rosser provided a form of G that is sufficient to invoke simple (negation) consistency. This necessitates a slightly different (thoguh equivalent) axiomatic treatment. Godel and Rosser both understood that the form of G that TVAA uses, the one that he gleaned from the mass of popular presentations and high-gloss websites, is not sufficient to escape omega-consistenct.
TVAA, though, assures us that it is sufficient. Morevover, he is going to supply a proof . . . maybe . . . if he can figure out what the word “formal” means.
:rolleyes:
The consistency of arithmetic can be proven. I told you that already, too, in one of the many posts that you have failed to read and understand. Specifically, Gentzen proved the consistency of a Peano Axiomatization in 1936 using transfinite induction. This means, of course, that if we accept the axioms of tranfinite deduction we can assert that a Peano Axiomatization is consistent. This is a theorem of transfinite induction. It has no implications for the consistency of a P.A. in other systems. I have seen this is described as: “assuming the dubious to prove the obvious”. Most folks just asusme that P. A. is consistent, but that is not the same thing as proving it.
(Note: the natural numbers can be proven consistent using only set theory, but while a P.A. almost defines the natural numbers, but the axioms also allow nonstandard models of arbitrarily large cardinality [in other words, uncountably infinite natural numbers])
:rolleyes:
You are an idiot.
That is Godel’s Second Theorem. I have specifically referenced that result, too. Why you think this somehow refutes anything that I have said about GIT is something that only you and the voices inside your head can answer.
:rolleyes:
You are an idiot. You latch upon the word “truth” but fail to read and comprehend the whole of the argument. From the cite I linked to:
[ul][li]The undecidability of some arithmetical propositions within the deductive system S may be classed among the syntactical metamathematical characteristics of the system S (represented by the calculus P), for the reason that this undecidability derives from the undecidability of some formulae within the calculus P which represents S. Deductive systems, unlike calculi, have also semantical metamathematical characteristics; in particular their propositions have or lack the semantical property of being true–what Gödel in his introductory Section I calls being “correct as regards content” (inhaltlich richtig). [/li]
Here, the author points out that the undecidability of G is a syntactic result that holds for calculi. He also notes that undecideability within a calculus pairs to a similar corrolary within a deductive system (Godel’s usage here is non-standard, BTW) In teh deductive system, we can speak of the semantical property Godel called “inhaltich richtig”, and which most folks gloss to “true”.
[li]Connecting the syntactical property of being provable with the semantical property of being true by taking every proposition provable within S (i.e. every axiom and theorem of S) to be true (see [176]) gives an additional kick to the undecidability in S of g–by adding that g is true. [/li]
Here, the author observes that if one adds the additional requirement of soundness (“every proposition provable within S to be true”). This is just an informational note, since GIT does not require soundness. That, too, would make GIT LESS GOOD.
[li]In his introductory Section 1 Gödel intermingles semantical with syntactical considerations in sketching a proof of the undecidability of g (which is the reason why I have seldom referred to this section in this Introduction). The distinction between what is syntactical and what semantical was not made explicitly until a year or two later (by Tarski, whose work included rigorously establishing unprovability theorems that were semantical); [/li]
Here the author notes that Godel’s treatment mingled semantic and syntactic concerns, but Tarski later eplicitely separated the two and focused upon semantical (“truth”) arguments.
[li]it is implicit in Gödel’s remark towards the end of Section I that “the exact statement of the proof [of the undecidability of g], which now follows, will have among others the task of substituting for the second of these assumptions [that every provable formula is also correct as regards content] a purely formal and much weaker one” [/li]
This is the author’s reference to Godel’s mention tha t"truth" should/could be dropped from a rigorous presentation of teh method that his paper supplies.
[li]Gödel’s proof in Section 2 is a purely syntactical proof about a calculus (the formal system P) whose interpretation as a deductive system for arithmetic is, strictly speaking, irrelevant to his argument. [/li]
Here the author explicitely states that the semantical element (“truth”) is irrelevant to a proof of GIT.
[li]It is true that Gödel explains arithmetization as a way of co-ordinating strings in his calculus with natural numbers, and he discusses recursive functions in terms of natural numbers (and I have followed him in speaking of numbers in both these contexts). But whenever he talks about numbers, and thus makes a remark which is prima facie about a deductive system rather than about a calculus, the remark is always a syntactical remark about the deductive system, and is therefore in essence a remark about the calculus which represents the system. [/li]
Here the author explains that, while Godel did intermingle (as mentined above) syntactic and semantic concerns, whenever he specificaly addresses the deductive system (“numbers” as opposed to “numerals”) his proof addresses only syntactic elements, and thus actually pertains to the calculus not the deductive system.
[li]For example, when Gödel says at the beginning of Section 2 that his formal system P has “numbers as individuals”, and speaks of “variables of first type (for individuals, i.e. natural numbers including 0)” [176], all that is relevant to his argument is that numerals are the only substitution values (not containing variables) permitted for his variables of first type. [/li]
Here the author gives an example to illustrate the case outlined above.
[/ul]
:rolleyes:
What I claim is true, as is demonstrated both in the links that you misread and in the logical arguments that you fail to refute. It is not the case that every system subject to GIT must necessarily include a binary truth operation that is defined for all well-formed formulas. Once again, such a restriction would make GIT, like “TIT”, LESS GOOD than the actual result.
:rolleyes:
You are an idiot. What Rosser did was find a better form for G and use a slightly different (but equivalent) axiomatization. Both of those things are within the intellectual grasp of any reasonably intelligent lay person with an interest in logic and mathematics. That you both fail to meet that standard and cannot be bothered to even investigate Rosser’s contribution to GIT speak for themselves.
:rolleyes:
:rolleyes:
The first statement is simply another TVAA lie.
The second statement is both a TVAA lie AND a prime example of TVAA ignorance. I never said that GIT only holds for omega-consistent systems. I said that the version of “G” thet you are using holds only for omega-consistent systems. There was no ambiguity in my statements to that effect, yet you still manage to misrepresent them.
:wally
The above represent my last words on this subject.
TVAA may, if he chooses, have as many last words as he wishes. I have every confidence that they will be as honest, insightful, and compelling as his nature allows.
Spiritus, I want to say that that link is one of the best fucking presentations of GIT I’ve read yet. though the internal links seem to be missing, if you look at the http:// address you’ll see x.html where x is a number. Replace that with 1, 2, etc, to get all sorts of cool shit, including the translation of the original paper.
How you found that I’ll never know, but it was so fucking clear it really helps me. Now, about Godel-Rosser… 
Hmm. I can’t get this cryptic crossword…
“Japanese airplane votes for TVAA”. Four letters, begins with a “z” and ends with “o”. {scratch head}. zapo, zaio, zuno, zygo…
Sigh. Maybe it’ll become clearer if I sleep on it.
Erl, the inernal links work if you go here.
I actually stumbled across ygdrassil while trying to hunt down some references for Finnish and Scandanavian myths/sagas. They don’t have the Kalevala, but they do have a version of Beowulf (though I have been spoiled for other versions since getting my hands on Seamus Heaney’s translation) and a great collection of Icelandic sagas. I found the sagas first, then hit the index of links and thought – WOW! The mother load. How can you not love a site that gives you Thucidides and Mark Twain, Sun Tzu and Kant, Tocqueville and Jules Verne.
And Godel, of course, a wonderful treatment of Godel.
Well… A little care needs to be taken before such enthusiastic words. There are many odd items on the list. Henry Ford’s The International Jew for example. Part of “the core literature of Western Civilisation”? And the introduction says
Be aware that this is a Nationalist site.
Hmmm – good catch. I never thought to check the site for political bias. Still, I don’t much care whether they present a balanced view of Western Civilization. It’s a fine collection of public domain documents in electronic form. Plus, one never knows when the need to read the Unibomber’s manifesto might suddenly arise.
Let’s review:
GIT itself doesn’t make any claims about the truth of falsehood of the statements it shows cannot be proven in a sufficiently powerful system.
GIT-2, however, shows that the consistency of a sufficiently powerful, consistent system cannot be proven within itself.
Since the system is consistent, the statement “this system is consistent” is true.
The statement “this system is consistent” is unprovable.
Therefore, the statement “this system is consistent” is guaranteed to be true but unprovable for all sufficiently powerful, consistent systems.
The versions of GIT apply to all consistent systems. This section of the debate began when Spiritus claimed that the version I referred to held only for omega-consistent systems.
Godel’s results have been demonstrated without omega-consistency.
More to the point, your source also clarifies just what ‘consistent’ means in regards to mathematics, and more specifically in discussions of GIT… and it’s freedom from contradiction.
As you’ve pointed out, Godel’s original results worked only for omega-consistency. Rosser was able to show that omega-consistency wasn’t necessary for GIT to hold… so how are you disproving my point, again?
Spiritus has repeatedly claimed that isn’t the way ‘consistency’ is used in mathematics, and that it has a much weaker definition. When asked to give evidence that this is the case, he linked to a site that defined several specific terms under ‘consistency’ but not the word itself.
Unless altered by a qualifier word or phrase, ‘consistency’ always refers to freedom from contradiction: A and ~A cannot both be proven in a consistent system.
What does Spiritus do when dictionaries and online discussions of GIT (including his own) clearly indicate that my use of the word ‘consistency’ is correct? He simply accuses me of lying. He does this repeatedly without ever demonstrating a source that directly contradicts this use or that proves his claimed definition is the correct one. (See above quote from his source)
What does any of this have to do with my point? (Absolutely nothing, which is why Spiritus brought it up in the first place.) I’m using that form of G only as an example. GIT-2 doesn’t depend on omega-consistency (which Spiritus has repeatedly ignored throughout this debate), and GIT itself required it only initially.
** Is this merely a lie about what I’m claiming, or does Spiritus simply misunderstand? I can’t tell – and I can’t tell which would be giving him the benefit of the doubt: assuming he’s stupid or assuming he’s a liar.
I’ve never claimed that the system GIT applies to must have binary truth operations that are defined for all well-formed formulas. There are obvious exceptions: “this statement is false” is neither true nor false, but something else. The statement “this system is consistent” does necessarily have a well-defined truth value. Either a system is consistent or inconsistent: there’re no known additional possibilities. For consistent systems, that statement is always true. For inconsistent systems, that statement is always false.
GIT doesn’t say much about what insufficiently powerful or inconsistent systems can do (if it’s not sufficiently powerful, we can’t generate proofs, and if it’s not consistent, there might be a proof of the statement “this statement cannot be proven”). It talks about sufficiently powerful and consistent systems – and within these systems, there are statements that can be made that are true but unprovable. Always. That’s what GIT-2 makes clear.
My sources indicate that Rosser was able to work with GIT without using omega-consistency. As you state, he found a better form of **G]/b]. How does this contradict anything I’ve said?
my cat mittens smells like an insufficiently consistent system of cat food
I like sheep
Look, folks, this is really simple:
First, Spiritus Mundi made a specific claim about the definition of consistency. He offered no sources to demonstrate he was correct or support his argument, and he responded to all requests to produce evidence in his favor with :rolleyes: or silence.
He was also wrong about that definition.
He is right that the definition of consistency (read: simple consistency) without a non-trivial definition of ‘truth’ isn’t enough to demonstrate the particular Godel statement I referenced. A system could be completely consistent (it’s impossible to prove the negation of all provable statements) while permitting untrue statements to be provable; for example, a consistent system could contain the statement “the system in which this statement occurs is inconsistent”.
The whole point is that when we include a non-trivial definition of truth, this problem vanishes: that’s what omega-consistency is all about. A consistent system with a meaningful definition of truth does allow Godel’s point to be demonstrated.
Spiritus doesn’t understand the meaning of the arguments he argues about so virulently; he disguises his lack of understanding with terminology and misdirection.
He never actually presented a counter-argument to my points, either.
TVAA, I should point out that it’s not YOU personally that is capable of fighting ignorance. It’s the process of discussion, argument, and criticism itself, given enough intelligent people with diverse views. Sure, some people can subvert discussions for a time with various diversionary tactics and quibbling. But in the end, consistent clear, well argued, and well supported arguments are going to win out. But you can’t just assume beforehand that those are your arguments. No one person gets to decide beforehand who’s smart and who’s ignorant: it’s just a general outcome of the process, not of any one person’s righteousness.
Spiritus is actively opposed to that discussion.
Most of the time, he responds to analysis of his arguments by saying something like “I thought you’d think that” or “that just what I’d expect you to think”.
He ignores the circumstances and context of the discussion to bring up technical terms whose definitions don’t actually change the nature of the debate.
He constantly made accusations that I was lying, mostly when I disupted his claims about the meaning of concepts…
It makes no difference whatsoever if I conclude Spiritus is ignorant – either he’s ignorant, or he’s not.
In any system where ‘truth’ has the generally-accepted meaning that disallows contradiction and is sufficiently powerful to allow conclusions to be drawn, Godel’s work shows that 1) there will always be certain statements that cannot be proven within the system, and that 2) some of these statements will be true relative to the system.
Computers aren’t merely devices made of silicon and metals: they’re abstractions too. There isn’t a meaningful way to distinguish between abstract and concrete systems. The distinction is ultimately arbitrary.
There are always ways to pose questions that specific computers can’t handle.
Whether two systems are similar or not has nothing to do with how we perceive them. Television signals preserve information about the electromagnetic waves that bounce from objects, but without a television it’s doubtful anyone would notice the embedded data.
I will not be called stupid, ignorant, and uneducated merely because I don’t do a good job of representing the truth. These are facts, not my own opinions.
And when my opponents can’t perceive variations in meaning, and can’t use standard language to explain their positions… let’s just say their claims become especially ironic.
Ironic indeed. You’re quite the greased cat.
I once defined goodness for you as the aesthetic most valued by God. I defined love as the facilitation of goodness. These clear and simple definitions did not satisfy you, and you took it upon yourself to submarine every discussion I attended with your distractions and diatribes. You demanded definitions of the words in definitions — and yet, when the same was demanded of you, you balked and bailed.
And even now, you think that Spiritus Mundi has failed to state his argument clearly while the truth is that you’re too stupid to understand what he is saying. There are nine pages (so far) in this thread of people telling you that you need to look again at what the problem is. Let me make it as simple as I can.
The problem is you.
Oh, please.
Defining ‘goodness’ as “the aesthetic most valued by God” is like defining ‘grobnorsi’ as “the color most liked by God”. It doesn’t let us determine what grobnorsi looks like, or what things are grobnorsi.
Your definitions provide virtually no meaningful data. If you have pre-established ideas of what God values and what love is, then you can imagine that they’re capable of permitting rational discussion and analysis of your assertions; otherwise, there’s not much we can conclude from them.
Frankly, given the quality of argumentation you’ve offered these boards, I don’t think you’re in any position to analyze or criticize anyone else’s – as you’ve amply demonstrated in the “Randi’s tests” threads.
Well, once again, that’s just how stupid you are. The attributes that qualify an aesthetic are not the same as the attributes that qualify a wavelength. This is just more of your famous and ironic equivocation. You use terms extremely loosely. Like “color”, for instance. And yet you demand precision from others beyond which language is even capable of providing, owing to the very nature of definitions, i.e., circularity.
I’ve defined goodness very precisely. You are simply incapable of comprehending either the definition or the context that makes the definition perfectly acceptable. You’re just doing both with me and with Spiritus what you always do — projecting your own intellectual weakness as the failing of someone else.
No one is suggesting that the attributes that define a wavelength are the same as those that define aesthetics, stupid. Stop bringing up strawmen.
Aesthetics still require definitions and well-defined attributes to be applied; to be understood, these definitions and attributes must be made explicit.
Your definition contains virtually no useful information.
I could define “goodness” as “that which Samantha Johnson of Des Louiser, South Dakota, prefers”. But unless we can ask her what her preferences are, or observe the choices she makes (and thus deduce those preferences), the definition doesn’t allow us to draw many conclusions. Could we determine whether something is good? No.
So ‘goodness’ is whatever God values most, yes? So what exactly is that? What are the values that God values most?
You continually make claims based not on reason but on your own convictions and prejudices, then you insist that others just aren’t intelligent enough to understand why those things must be true. The problem is that you simply cannot imagine that those things might not be true – and since you can’t even consider the possibility, you begin reaching for ways to justify your convictions.
There’s a reason your “model proof of God’s existence” is a running joke on these boards, Libertarian, and it’s not because your insight places you beyond our conception.
grobnorsi? seriously what planet is the guy on?
Has anyone tried those new Pop Tarts Snak-Stix? 'Cuz Pop Tarts are a damn fine snack, so I was thinking maybe I should try the Snak-Stix, but then why mess with perfection?
My favorite Pop Tarts flavor is Cinnamon & Brown Sugar.