The problem with this statement is that is false. I have already given an example of a theory with a statement that has been shown to be undecidable.
For everybody else that is getting cross-eyed, there is Gödel for dummies: 
http://members.rediff.com/TheOne/godel.html
More explanations to fry your brain, if it is not there yet, here:
http://www.earlham.edu/~peters/courses/logsys/g-proof.htm
I appreciate the links, and will examine them in an attempt to fry my brain shortly. I’m curious though, what you presonally think is wrong with the idea that if undecidability necessarily is always undecidable, then how does it redeem the interpretation of undecidability from this theorem?
Here’s a fellow who goes to all this trouble to plug self referential statements into mathematics, but seems to miss the most obvious one…
This statement is undecidable.
What on earth could the concept of undecidability be referring to if it’s not referring to a claimed (or ‘proven’) indeterminability within the scope of thought or logic?
Undecidability is necessarily referencing the concept of undeterminability.
Even the slightest inkling of assent to the existence of undeterminability must necessarily effect all possible “levels” of all possible “systems”. To introduce the concept of undeterminability, effectively means, undeterminability EXISTS.
This means that an aspect of existence is not bound by any possible inference or causality or axiom. Which means that the claim of uncertainty is itself uncertain.
How is that a wrong thing to say or conclude? You may Believe that what you are seeing is proof of a necessarily decided undecidability, but what I’m saying is that it doesn’t make sense if you assent to the existence of undeterminability in order to make the claim.
What does undecidability MEAN if not used in reference to the asseent of the existence of undeterminability? Just think about it. Can you think of a method of detaching this relationship?
justhink… is that you?
Isn’t it required that the axiomatic system be consistent? And isn’t the definition of consistency that a statement cannot be proved true and false? I thought Godel constructed a statement that said of itself that it could not be proved nor disproved? (How that was done I don’t know).
Hmm, let’s start by assuming that a slightly different approach will help make ultrafilter’s quite correct explanation of indeterminism more clear. Before we get to Godel, let’s start with a trivial logical system. The only rule of inference is modus ponens (if P then Q inferences). The complete set of symbols is: P, Q, R. The complete set of axioms is:
[ul][li]P[/li][li]P -> Q[/ul][/li]Now. What is the truth value, in this system, of R?
That’s right . . . it cannot be decided. Yet R is a perfectly valid statement in our system. Notice, please, that indeterminacy does not require an ability for self-reference. That was just the means that godel used to demonstrate the incompleteness of certain formal logics much more powerful than the trivial one above.
This statemetn is simply false. The indeterminability of R within the simple system above has no consequences for teh inference set. It does not collapse to zero. It remains what it is: a perfectly fine little system that cannot tell us whether R is true.
You have some other implications of Godel’s theorem wrong, but I don’t think much will be gained by dwelling on them. You have clearly stated that your target is not GIT itself but all ideations of undecidability. So, rather than get dragged too far into formal logc how about if we concentrate on the conceptual base?
Yes, it is. This is not a problem for most people. Perhaps you could explain, in small steps rather than leaps of insight, why you think it should be a cause for consternation.
I believe you are wrong. There is no causal implication between a human mind’s ability to manipulate an idea and the existence of an objective instantiation of that idea.
Of course, it is also true that indeterminacy does exist for man, many formal systems (and informal) systems. So what? Why do you think that the existence of indeterminacy prevents us from saying anything definite about indeterminacy? the existence of falsehood does not prevent us from saying true things about lies.
No, it does not.
It means that many statements exist that cannot be assigned a truth value from within the system in which they are expressed. That is all. We are not prevented from saying definite things about R, or about a Godel statement, or about indeterminacy itself.
Not at all. I am absolutely certain that the truth value of R cannot be determined in my little system. I am equally certain that in formal systems subject to GIT there are statements that cannot be proved and cannot be disproved.
Likewise, it is a true statement that some statements are lies.
I will concede, ultrafilter, that I oversimplified the case of Goedel’s statements, perhaps to too great a degree. The quote GIGOBuster provided explains the situation more thoroughly.
Pedro, the whole point of Goedel’s work was that in 1900, mathematician David Hilbert called for an axiomatic system that would be able to generate all true statements of number theory. He believed a system could be created that was complete and consistent (i.e., the system would generate ALL possible true (truth value=1) statements, and generated NO false (truth value = 0) statements).
Goedel created a mathematical statement along the lines of “This sentence is false”. It in essence stated “This statement is not derivable within the axiomatic system at hand”. If you were to actaully derive this statement within the system then, you would have proved its truth value is 0, thereby wrecking your system by deriving a false statement. If you were to prove somehow that it were not derivable, then it would be a true statement that your system could not generate, also wrecking it. Such a statement can be written in whatever axiomatic system you care to create. The upshot is that an axiomatic system may be complete, or may be consistent, but CAN NOT be both at the same time.
I see, however, that our OP’er is after bigger fish.
I feel, in essence, olanv, that you are using two-state logic (T or F) to examine a system with three states (T, F, or N , where N means “undetermined”) or more (see the work of Lofti Zadeh, et. al.).
Within a two-state system, such as classic predicate logic, you can prove a statement true or false only (i.e., that a statement’s truth value will always be 1, or always 0, although Goedel showed your system of proof may not work all the time). In a three-state system, you can prove truth, falsehood, or undeterminability (i.e., that a statement’s truth value will always be, let’s say, 0.5).
You claim to be after a proof of the determinability of God’s existence, which you seem (to me) to be pursuing by saying that undeterminability does not exist. Within a two-state system, this is correct. Within a broader system (three-state, or fuzzy, for example), it is most certainly not correct.
Unless you are prepared, here and now, to present rigorous irrefutable proof fo God’s existence or non-existence, then the truth-value of the statement “God exists” is neither 1 nor 0, and if you wish to continue with the topic at all, you must acknowledge that we are discussing this in a system of more than two states. Let’s assume three states, and acknowledge that, at the moment, “God exists” has a truth-value of 0.5, or N, or “undetermined”.
Now, just as no one as proved the value of the statement to forever be 1 or 0, I am not aware of anyone ever presenting a rigorous proof that it shall forever be 0.5, which is the equivalent of saying God’s existence is “undeterminable”. You wish to discover a proof that the value can NEVER be 0.5, which I refute by saying that that is exactly its value at the current time.
If you wish to claim that the entire universe exists in a two-state system, I will tell you that I stand roughly 5’11", and ask you to assign an irrefutable truth value of 1 or 0 to the statement “scotandrsn is tall”. I suggest you read Lofti Zadeh’ papers on fuzzy logic.
You will have to be content with the fact that since proof does not exist for the statement “God exists” having the unchanging value of 0.5, that there continues to exist a possibility that a proof will emerge of its having a value of 1 or 0 for all time.
I should have made clear that I did not mean to imply that you can ALWAYS prove truth values for ALL statements, just that the framework of the logic system dictates what truth values can be proven.
Let’s throw out some vague notion of “Rules” as in, “Rules allow me to conclude that a proposition cannot be decided.”.
How are you proving this one way or the other? I want to see what claims you’re able to make about “no rules” – because you’re making the claim that “rules” allow for undecidability – which is basically, “using rules to show that the rules converge and collapse at certain points, to which the proposition cannot be determined as anything but undecidable within it’s own rule system.”.
Well, how do you know that your rule system isn’t “undecidable” as well?
From what vantage point are you getting a clear veiw of this horizon? Have you delved into the deep complexities of the determined undetermined undeterminism of “no rule land” from which to falsify this claim? Are we supposed to take your word that you have somehow been to and emerged from the abyss, conferred with “no rule land” and have been assured that your system of rules isn’t vulnerable to undecidability, and that the “lands of no rules” has decided to give permission to the masses that it does occupy certain territory, but says that it’s won’t invade any other territory if you just believe the WORD from the abyss? What is “higher order logic” if not some vague deity from on high to which a person takes pride in their ownership of truth, by stating that “It’s undecidable in the depths where it’s simple, but up here in this vague land of people who have traversed and emerged from “no rule land”, we have an abstract understanding of what is true, an enlightened veiw.”.
Dontcha kinda think there’s a conflict of interest here with the motive… I mean talk about job security for just about any organization of fraud on earth… “Undecidability exists as a concept that does not refute itself… and here’s why…”. I would say that in a society where fraud occurs on a regular basis, it would be a miracle if such a ‘proof’ wasn’t heavily funded and publisized and held at the heights of higher learning as the peek understanding of erudition. “Hmm… you’re mind is too simple to understand the revelation from the abyss, take your penance.”. “Undecidability as a conclusion that “rules can allow for the conclusion of undecidability” does not in anyway render the rules we used to arrive at this certainty as subject to undecidability itself – just because propositions CAN be undecidable, doesn’t mean that OURS ARE, and we know this, because we used OUR rules to decide that undecidability exists!! Bow to the abyss from which I spawn the knowledge of undecidable creation!!”
From what is this falsified AGAINST?
Let’s look at this from a conservation of matter point of veiw.
You have decidable propositions from which rules allow a person to conclude that the proposition cannot be decided.
How do we prove that undecidability itself doesn’t consume the entire set of rules, if we have determined that at least one set of rules allows for undecidability to be concluded as not self refuting?
Where is the rigorous proof that “No Rules” allows for decidable propositions, undecidable proposition, propositions at all, no propositions at all? How does one demonstrate “no rules” from which to falsify this? How does one then show some sort of mechanism from which “rules / no rules” operates on decidability if they’ve already shown that “rules” allows for the conclusion that a proposition cannot be decided?
So what do you say about this plane of “no rules”?
Can decidability emerge from “no rules”?
Can undecidability emerge from “no rules”?
How are you proving this one way or the other? I want to see what claims you’re able to make about “no rules” – because you’re making the claim that “rules” allow for undecidability – which is basically, “using rules to show that the rules converge and collapse at certain points, to which the proposition cannot be determined as anything but undecidable within it’s own rule system.”.
But, think about this. If the rules collapse in one system, then it begs the question of whether the rules collapse in all systems.
Maybe this is just a semantics game.
What’s the difference in peoples minds of “undeterminability” and “non-determinability”? And even more, what does it mean to “determine undeterminability”?
Or even more, if this actually becomes a relevant argument, then the powers that be will need to swoop in and create the term “adeterminability” as a means of keeping their concept in the meme pool, not to protect truth, but to protect the fact that their vital interests are dependant upon and have only stemmed from falshood.
So, I want to hear it.
Define:
Undeterminability
Non-Determinability
Adeterminability
Anti-determinability
Indeterminability
In my book, when you negate determinability (not determined), and declare that this negation EXISTS as something that’s not self refuting, you are opening up a system wide logical vaccuum. To emerge from the “abyss” of the “learned text” and profess your ability to determine undeterminability as exiting but not self refuting – is nothing short of the attempt to protect, to secure, to fight for, to live for and to die for ambiguity. Not because it’s not self refuting, and this truth actually does exist, but because ones very life force is drawn to protect the thing that will necessarily kill them – because it allows them to live a short span of time as a king over others who try to dismantle this ambiguity in order to redeem life for all being.
I’m not envisioning this concept of “undecidability” as having limited application in an existence where things are contingent – I see it as a pattern of authority attempting to confuse truth and fight for ambiguity; it’s as old as the Bible, it’s older than the Bible. Ambiguity gives you your power over others and then takes all life away – and in your small mind, this glimmer of power over others is more important than all the effort used to try to secure your permenant place in life should you so desire - because when push comes to shove, that power only came from attacking them, mocking them, kicking them - not for reasons true, but for reasons shrouded in ambiguity – each action being the intentful defense of ambiguity, to feed its strength; the strength of the only thing capable of destroying you. You fight against people who don’t kick, who don’t mock, who don’t attack – people who cannot live for untrue reasons… people who fight for your survival, and in your swelling circular ego, this to you is proof of life, meaning and victory, the mantra, “Self refutation is truth.”.
It may raise the question (begging the question is different entirely), but it’s an easy question to answer. In a system where the only symbol is P and P is taken as an axiom, every sentence is decidable.
It’s starting to look like you’re not actually interested in learning, but that you just want us to declare you right. I wouldn’t hold your breath on that…
I will happily attempt to answer your latest set of questions, but only if you will kindly address the points that have already been raised in response to your “argument”.
In some circles, such an exchange is known as polite discourse.
I should ammend: *I will happily attempt to answer your latest set of questions except for the bizarre fantasy that some powerful “organizations of fraud” are and have funded the publication of uncertainty theorems in some incomprehensible scheme to amass greater wealth and power.
:rolleyes:
It is far, far easier to defraud someone by telling them that you have the answer than it is by telling them that the answer cannot be had.
In a system where the only symbol is P, P is the only thing that exists. This means that P would be nothing. It’s a singularity without reference (which means that it cannot even be a singularity).
You’re declaring the existence of something that by definition must be nothing, and declaring that your ability to assert this existence is not self refuting. It’s very similar to the argument I’m leveraging against the concept of uncertainty. The whole point of logic is to avoid situations where you can make linguistic representations that collapse the definition.
-All grass is green
-I am red grass
I am green
The argument,
-All grass is green
-I am red grass
I am green
Is the same type of argument being used by you, when stating that all symbols are equivicated when the symbols in the theorem itself are not all equivicated. What does it MEAN to say that all symbols are equivicated?
The statement is referencing itself. It’s as if you have secret commune with the “land of no rules”, and from this ‘empass’ you declare existences of validity from nothing - as if you have an inside track on what “nothing” TELLS you that it occurpies and doesn’t occupy. That you’re this authority on nothing, and as a logitician, are given this priviledged commune - a prophet.
I don’t know, because I have never said that.
What does it MEAN to say that P is the ONLY symbol and is taken as an axiom. Now you’re just playing cat and mouse with the point I’m making.
“in a universe of discourse” is implied. You cannot IMPLY a universe of discourse in a universe where nothing can be discoursed. You know that this is an assertation of a singularity, you’re just playing cat and mouse at this point.
Four “theories” of contention, for which there is no evidence…
Undecidability
Undeterminability
Singularity
Freewill
In this instance you’re invoking the singularity to prove your point about the undeterminability. They are not only question begging, they are the BIGGEST question begging questions in the history of this planet!!
If we take a formal system to be a 4-tuple (L, A, R, T), with L the language, A the axioms, R the rules of inference, and the set of permissible truth values, then I have constructed a formal system with L = {P}, A = {P}, R = {P}, and T = {0, 1}. Could it be any clearer?