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olanv
01-14-2004, 10:13 PM
The claim of any undeterminability in existence necessitates that the stability of the claims truth cannot be determined. Thus the purpose for asserting the claim as true is self refuting.

Firstly, I'm willing to argue the point.
Secondly, to address (hopefully) whatever ambiguity remains of this statement, I'll explain further.

Let's say you have a logical formulation that concludes undecidability. Undecidability, is a method of expressiong indeterminability within the 'sphere' of thought. Although, granted, it could leave open that there is still an operative causality - just that it's not accessable to thought.

But, this interpretation, I believe is false, for it is still declaring an indeterminancy in existence - as thought itself exists.

The key here, is that by declaring an undecidability within thought, one is declaring an undeterminability within existence. This is contingent upon the acceptance that the perception of thought, and the perception of a conclusion of undeterminancy in this perception of thought both exist.

So, in this instance one is claiming either that existence itself is undeterminable or that something is undeterminable within existence (a perception of a perception of thought would be what exists in this instance, that is used to "observe" the belief of an undeterminancy conclusion -- so as to humor the solipsists).

The problem, and the refutation, emerges when one realizes that they have consigned themselves, or rather, assented to the existence of undeterminancy as necessary, even if it is only necessary as a conclusion that follows from some abstract premises that themselves may not even be true or valid.

Again, the important part, is that it has been accepted in some way, that undeterminancy exists.

The problem, is that if there is actually an aspect of existence (or rather existence itself) that is undetermined, then there is a logical backhole that necessarily exists from which the stability of the statements' truth value cannot be determined. The very act of asserting the truth value of the statements' undeterminiability is false.

The conclusion is that statements cannot have undeterminancy as a conclusion without refuting the act of making the claim on the part of the person who concludes such an answer.

olanv
01-14-2004, 10:26 PM
The claim of any undeterminability in existence necessitates that the stability of the claims truth cannot be determined. Thus the purpose for asserting the claim as true is self refuting.

Small error, big difference.

Thus the purpose for asserting the claim as true undeterminable is self refuting.

Rather, it is to say that if undeterminability is either determined or undetermined, it must be determined either way.

I'm not sure how to logically represent a claim. I tried playing with symbols, but discovered that I really need a macintosh for this for whatever reason, unless I make up my own symbols and translate them.

olanv
01-15-2004, 01:57 PM
Maybe someone would reply if it was suggested that this concept refutes the concept of undecidability in general? Like Godel's "Undecidability of certain propositions". I'm just not sure how one would represent such concepts logically. The argument seems solid enough.

Trinopus
01-15-2004, 02:01 PM
Mebbe I'm wrong, but as I understand it, Goedel proved that there are statements that are undecideable...but he didn't give an example...

*Can* you give an example? Can you produce a statement, X, and say, "I can prove that statement X is undecideable?"

It seems to me that this is the same as proving that statement X is false.

(To use the Turing Machine "halting problem" as an example, if I can state, definitely, "This program will never halt," that's the same as saying the statement is false. It's only those cases where we don't *know* if the machine will halt or not that are undecideable.)

Or...have I just played Bishop to King's Bishop Three...while everyone else is playing Bridge?

Trinopus

olanv
01-15-2004, 02:23 PM
Mebbe I'm wrong, but as I understand it, Goedel proved that there are statements that are undecideable...but he didn't give an example...

*Can* you give an example? Can you produce a statement, X, and say, "I can prove that statement X is undecideable?"

It seems to me that this is the same as proving that statement X is false.

(To use the Turing Machine "halting problem" as an example, if I can state, definitely, "This program will never halt," that's the same as saying the statement is false. It's only those cases where we don't *know* if the machine will halt or not that are undecideable.)

Or...have I just played Bishop to King's Bishop Three...while everyone else is playing Bridge?

Trinopus

My point here is that what I'm talking about refutes the conclusion of undecidability. You're asking me to produce a statement of undecidability, when the proposition explicity articulates that one connat be rationally made.

The argument is basically this.

This proposition exists.
This proposition is undecidable.

If it's undecidadable, then it's possible that the proposition does not exist or is NOT undecidable. Basically, it states that a proposition cannot be undecidable.

I covered this earlier by stating that undecidability is defined as indeterminability within the scope of thought.

If you agree that the proposition exists, and you agree that the thought to interpret the proposition exists, and you agree that undecidability is defined as indeterminability within the scope of thought, then it follows that any undecidability theorem is an undeterminability theorem.

This connection is made through the defition of undecidability as, "indeterminability within the scope of thought", and the premise that thought exists. It follows that undeterminability must exist then, should all of these premises be accepted. When one accepts that undeterminability exists, then they are necessarily accepting that the truth of deciding upon undecidability is is accepting the truth of indeterminability within existence.

This effectively means that there is an aspect of existence that cannot be causal, or determined. This creates a backdoor from which it becomes inconsistent to agree with your own conclusion of undecidability, as such a conclusion is not stable in an existence where undeterminability exists.

If you decide that the proposition doesn't exist, then there's no reason to interpret it.
If you decide that the proposition does exist, there there's no reason to interpret it as being undecidable.

ultrafilter
01-16-2004, 12:27 PM
Gödel's method of representing undecidable propositions relies on a computable bijection between statements in predicate calculus and numbers. By doing this, he was able to take statements about number theory (like "There is a proof of statement S in theory K") and turn them into complicated arithmetical relations. He then performed proofs about those.

olanv
01-16-2004, 02:36 PM
Gödel's method of representing undecidable propositions relies on a computable bijection between statements in predicate calculus and numbers. By doing this, he was able to take statements about number theory (like "There is a proof of statement S in theory K") and turn them into complicated arithmetical relations. He then performed proofs about those.

Ultrafilter,
Maybe you can help me here. I'm trying to figure out if this general thought process refutes Godel's Theorem. In saying this, I'm not saying that I know HOW to precisely refute his specific method of arriving at this conclusion, only that the conclusion of undecidability is irrational. It's a refutation by proxy. Somewhere, somehow, it is been concluded through Godel's Theorem that undecidability has occured, is occurring, does occur etc...

Methodology aside, it's the very claim that something can be reasonably interpreted as undecidable that's being argued here.

If you're stating that something is undecidable within a defined set of conditions, what you're saying is that undecidability exists!

This is where the refutation moves in and states that undecidability is defined as "undeterminability within the scope of thought". Something incapable of determination is something that is not subject to laws of cause and effect.
By stating that undecidability exists, one is stating that the definition of undecidability exists. Since undeterminability is in the definition of undecidability, it follows that undeterminability exists. Again, it's not important how undeterminibility is said to exist, just that it is explitly stated that it does in fact exist somewhere along the line of reasoning.

Now, if undeterminability exists, then the whole point for making a conclusion of undecidability refutes itself! The argument here is that it is always false to conclude undecidability as an interpretation from any argument.

ultrafilter
01-16-2004, 02:41 PM
Nothing can refute Gödel's theorem, because it's true. In fact, there are examples of undecidable statements in number theory--the simplest is that there is some statement that is not a theorem. This can be expressed as an arithmetical statement, but no proof is possible.

scotandrsn
01-16-2004, 02:52 PM
Ultrafilter,
Maybe you can help me here. I'm trying to figure out if this general thought process refutes Godel's Theorem. In saying this, I'm not saying that I know HOW to precisely refute his specific method of arriving at this conclusion, only that the conclusion of undecidability is irrational. It's a refutation by proxy. Somewhere, somehow, it is been concluded through Godel's Theorem that undecidability has occured, is occurring, does occur etc...

Methodology aside, it's the very claim that something can be reasonably interpreted as undecidable that's being argued here.

If you're stating that something is undecidable within a defined set of conditions, what you're saying is that undecidability exists!

This is where the refutation moves in and states that undecidability is defined as "undeterminability within the scope of thought". Something incapable of determination is something that is not subject to laws of cause and effect.
By stating that undecidability exists, one is stating that the definition of undecidability exists. Since undeterminability is in the definition of undecidability, it follows that undeterminability exists. Again, it's not important how undeterminibility is said to exist, just that it is explitly stated that it does in fact exist somewhere along the line of reasoning.

Now, if undeterminability exists, then the whole point for making a conclusion of undecidability refutes itself! The argument here is that it is always false to conclude undecidability as an interpretation from any argument.


What Goedel proved, to expand on Ultrafilter's posts, is that within the system of Prinicipia Mathematica, with axioms and precise rules for deriving new statements of valid number theory from those axioms, you can generate a statement and its negation through the valid use of those axioms and rules.

In other words, you declare that within the system, the axioms have a truth value of 1, and that any statement that can be generated through proper use of the rules also has a truth value of 1. Using the axioms and rules, as proved rigorously by Goedel, you can, for some statement x, create the statements "It is true that x" and "It is not true that x", both with truth value 1. We must conclude from this contradiction that the truth value of x is undecidable.

Furthermore, the nature of Goedel's proof made it clear that this was not a special case of Prinicipia Mathematica, but a property of ALL axiomatic systems.

ultrafilter
01-16-2004, 02:56 PM
Not quite--I don't have time to go into it right now, but I will later.

olanv
01-16-2004, 03:17 PM
Nothing can refute Gödel's theorem, because it's true. In fact, there are examples of undecidable statements in number theory--the simplest is that there is some statement that is not a theorem. This can be expressed as an arithmetical statement, but no proof is possible.

This is the point of contention.

"no proof is possible" because it is proven that it is unable to be determined through the use of thought (the interpretive mechanism) or logic (the representative mechanism).

By default, the contention here is that undeterminability exists!!

While it could conceievably be the case that the conclusion of undecidability does stand for all time given the condition of undeterminability (by chance the proof holds, but not because of any causal system implicit in the proof) -- the actual act of interpreting the proof as necessarily undecidable is self refuting. Do you understand what I'm saying by that?

"Within the scope of thought this cannot be determined" = Undecidability

By default, this means that undeterminability EXISTS!!

Which means that the very framework which is used to hold the truth value stable, is discarded. Which in effect makes the declaration of, or interpretation of undecidability self refuting.

ultrafilter
01-16-2004, 03:33 PM
Your argument ignores the distinction between proofs and metaproofs. Any formal theory has two languages--the language of the theory itself, and the metalanguage used to describe the theory. It's quite all right to miss this distinction, as it's not generally taught until you get to advanced logic classes.

Anyway, the undecidable statements are statements of the theory, written in the language of that theory. The statement that undecidable statements exist is a metastatement, written in the metalanguage of the theory. So it's perfectly consistent that undecidable statements exist.

olanv
01-16-2004, 03:42 PM
I'll give it one more try at conciseness, if you please.

If you have an aspect of existence to which you assent is undetermined, you are effectively discarding both the representation and interpretation mechanisms in one fell swoop, at least for this aspect of existence -- which incidentally, must hold for all of existence.. because what emerges FROM here (this undeterminable pocket) can change anything and/or everything anytime.

This is where the problem emerges. because what can emerge from this pocket of undeterminability is complete determinism, which effectively would refute the proof, because there no longer exists undeterminability. Not only that, but determinability does not revert back to undeterminability.

The shortest method of articulating the self refutation is that you're deciding undecidability (undeterminability) - which is an explicit statement of "It may be possible that this conclusion of undecidability will CONVERT to decidability through this uncausal mechanism, to which it will never again REVERT!! That being the case, it is illogical to interpret the actual theory as undecidable.".

What it's saying, is that the act of making the conclusion of undecidability only makes sense if you don't make that claim, or even further, if you don't make that interpretation.

The act of making the claim, refutes the conclusion.

It's not so much Godel that I'm after, although, this would certainly directly effect ALL claims of undecidability, but it also gets to the heart of one of the strongest undecidability claims made by both theists, agnostics and atheists alike -- namely that the proposition of God cannot be decided.

If this argument holds, then it can be shown once and for all, that such a claim is neither valid nor sound. This then places theistic arguements squarely on the cutting block -- yes, no -- no in betweens, no circumventions.

ultrafilter
01-16-2004, 03:47 PM
There's a book called Godel's Proof that does a decent job of introducing this material to the layperson. I think you would find it interesting, and that seeing how this sort of stuff is actually done would answer a lot of your objections.

I'm trying to put this as constructively as possible, but there are large parts of your argument that aren't even coherent given the standard meanings of several of the words here.

It's a fascinating topic, and I encourage you to study it if you're interested. I, and several others on this board, will be more than willing to help you work it out, but we need a common vocabulary and set of ideas.

olanv
01-16-2004, 03:48 PM
Your argument ignores the distinction between proofs and metaproofs. Any formal theory has two languages--the language of the theory itself, and the metalanguage used to describe the theory. It's quite all right to miss this distinction, as it's not generally taught until you get to advanced logic classes.

Anyway, the undecidable statements are statements of the theory, written in the language of that theory. The statement that undecidable statements exist is a metastatement, written in the metalanguage of the theory. So it's perfectly consistent that undecidable statements exist.

This is where we're crossing wires. I'm arguing that undecidability always refers to the meta system. When you state that undecidability exist IN THOUGHT -- you're talking about a sub system of the meta-system. However, the definition of undecidability refers to the existence of undeterminability within the meta-system in order to be distinguished. When you declare undeterminability within the meta-system, your discarding the stability of the claim in BOTH systems -- which effectively means that the stability of undecidability ITSELF is not rendered to causality -- which means that the interpretation of undecidability is undecidable!! From this we can determine that it's a contradiction to state that you have decided undecidability -- rather, such a claim produces a stasis that doesn't allow the interpreter to express the claim!! The moment the claim of undecidability is expressed, THEN, the statement becomes self refuting!!

ultrafilter
01-16-2004, 03:54 PM
It's not so much Godel that I'm after, although, this would certainly directly effect ALL claims of undecidability, but it also gets to the heart of one of the strongest undecidability claims made by both theists, agnostics and atheists alike -- namely that the proposition of God cannot be decided.

If this argument holds, then it can be shown once and for all, that such a claim is neither valid nor sound. This then places theistic arguements squarely on the cutting block -- yes, no -- no in betweens, no circumventions.

OK, I think I understand where you're coming from now. There's a difference between the informal concept of undecidability you've got in mind, and the very formal concept that I have.

You're still running into trouble due to the language/metalanguage distinction that I mentioned earlier, but in a way you're also barking up the wrong tree. The existing undecidability theorems apply only to theories about arithmetic, which are not the ground of theistic arguments.

olanv
01-16-2004, 04:06 PM
OK, I think I understand where you're coming from now. There's a difference between the informal concept of undecidability you've got in mind, and the very formal concept that I have.

You're still running into trouble due to the language/metalanguage distinction that I mentioned earlier, but in a way you're also barking up the wrong tree. The existing undecidability theorems apply only to theories about arithmetic, which are not the ground of theistic arguments.

Thank you so much for your time ultrafilter.

What I'm trying to articulate is that a claim of undecidability reverberates through all systems no matter which system it's in. If it's in the meta-system then it reverberates into both systems. If it's in the standard sysem, it reverberates into both systems. This is because the existence of undecidability opens up a "black hole" to which determinism CAN effect both systems. What I'm saying is that undecidability, as a result of this, is undecidable!! This isn't a problem until you make the claim of deciding undecidability... once this claim is made, a self refutation emerges.

This doesn't only effect the claim, this also effects the interpretation. If you interpret undecidability in a situation where my argument holds, then undecidability is undecidable, which means that the interpretation is false... BUT only when you make that interpretation.

Maybe it helps to state that I'm attaching as necessary that any claim of undecidability must be universal -- it reverberates universally, no matter what system it's being declared from or in. As a result, undecidability ITSELF is undecidable. Which means that the claim of deciding an undecidability is self refuting. This certainly does grasp out towards Godel, because Godel is making a claim of deciding an undecidability.

olanv
01-16-2004, 04:20 PM
OK, I think I understand where you're coming from now. There's a difference between the informal concept of undecidability you've got in mind, and the very formal concept that I have.

This is just ignoring my argument by appeal to authority. You are more then welcome to define your "very formal concept of undecidability", and compare it to what you see as my "informal concept of undecidability". Until this is resolved, we're playing an appeal to authority cat and mouse game.

I'm defining undecidability as "the conclusion that a proposition is unable to be determined within the scope of thought or logic.".

I certainly hope that you are able to define the "very formal concept of undecidability" in a similar manner, without the use of logical symbols and such, just a plain english statement.

ultrafilter
01-16-2004, 04:40 PM
This is just ignoring my argument by appeal to authority. You are more then welcome to define your "very formal concept of undecidability", and compare it to what you see as my "informal concept of undecidability". Until this is resolved, we're playing an appeal to authority cat and mouse game.

You do know that appealing to authority is only fallacious when the authority in question is no more qualified to judge the topic than the average person, right? The mathematical community is a perfectly good authority on the standard usage of mathematical terms.

I'm defining undecidability as "the conclusion that a proposition is unable to be determined within the scope of thought or logic.".

I certainly hope that you are able to define the "very formal concept of undecidability" in a similar manner, without the use of logical symbols and such, just a plain english statement.

If it were reasonable to do so, we wouldn't need to resort to the formal symbols in the first place.

That said, we need to define a few terms first. We will take "sentence" to be an undefined term, just because defining it does require a departure from ordinary English. For the same reason, we will take "language" to be undefined.

A rule of inference is a well-defined method for generating new sentences from old ones with the property that if the old statements are true, the new ones are as well.

A theory is a collection of sentences (called the axioms) and a non-empty collection of rules of inference.

A proof of a sentence S in a theory K is a sequence of statements ending with S. We also require that each statement be either an axiom of K or the consequence of applying one of the rules of inference of K to the preceding statements.

To wit, GIT states that if a theory K meets certain criteria, then there is either a statement S in the language of K such that there is no proof of S or its negation in K, or every statement in the language of K has a proof in K.

If a statement S in the language of a theory K has no proof in K, and its negation has no proof in K, then it is said to be undecidable relative to K. Note that a statement undecidable relative to K may become decidable if you add new axioms, or it may be decidable outside of K (i.e., when expressed as a sentence in the metalanguage of K).

olanv
01-16-2004, 05:08 PM
If it can be shown that undecidability in any system must refer to undetermination in all systems, and then that this means that undecidability itself is undecidable, and if this means that deciding an undecidability is self refuting when it's already been shown that undecidability is undecidable...

then wouldn't that effectively work as a refutation of any undecidability claim?

Maybe you're not quite grasping what I think is able to be done here.
I'm suggesting a complete refutation to the entire CONCEPT of undecidability; that undecidability always refutes itself.

If you actually get caught into the 'black hole' of this self referential infinite regress of undecidability, your mind will lock. The only way out of this without refuting yourself, is to ASSUME that all statements must be decidable. Which is to conclude that it is necessary that any statement deciding undecidability must have an incomplete axiom set, and or incomplete rules of inference.

To prove that these must always be incomplete, is to fall back upon undecidability, which again, is self refuting.

You're saying that it's reasonable in some way to decide undecidability. I'm saying that this is always a contradiction, but even more, a self refutation!

I'm not sure how much further we can get until you understand even 'abstractly', outside of your formal understanding, how I would even think such a thing.

Remember, I'm saying that undecidability is always necessarily undecidabile itself. Which makes the interpretation of something as undecidable, self refuting. I'm not just getting to any contradiction in the symbols or in making the claim... I'm actually trying to invade the heart of interpreting the claim.

I'm saying that it's self refuting instantly, the moment when someone interprets that undecidability is conceivably occurring.

This is because I'm positing that undecidability is necessarily undecidable, given that it's referring to the concept of undeterminability. In referring to the concept of undeterminability, I'm stating that the person is assenting to the existence of undeterminability.

It doesn't matter "only in relation to k", because what's being argued, is that the assent of undeterminability is the assent to something that circumvents the entire axiomatic process and the inference process. It would make inference a zero set, which you already said it cannot be.

Maybe you'd be willing to discuss on yahoo at some point? I'm not sure how much interested you are in this anymore, as you could be thinking that I'm simply too confused to comprehend what an argument is... but maybe this would help. Regardless, I do appreciate yor input. I think that this argument has potential for impact on what is argumentitavly sound. If the concept of undecidability itself can be refuted, that seems substantial to me.

ultrafilter
01-16-2004, 05:17 PM
Remember, I'm saying that undecidability is always necessarily undecidabile itself.

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.

GIGObuster
01-16-2004, 05:23 PM
For everybody else that is getting cross-eyed, there is Gödel for dummies: :)
http://members.rediff.com/TheOne/godel.html
For a general audience it might make more sense to state the theorem something like this:
"When the idea of paradoxical statements like "This statement is not true" was applied to mathematics, Godel was able to prove that similar paradoxes are found even in the most rigorous mathematical systems."
This provides some context for us to understand what the general problem for human reason is, and it give a more accurate picture of what Godel's theorem adds to the big picture of human reason.
Why bring up Godel's theorem at all? Unfortunately, the complexity of Godel's theorem makes it a cheap shot to use it to impugn the efficacy of reason, since it is unlikely that anyone will try to refute the assertion. I hope that a relatively simple analysis, such as the attempt above, can be developed to counter such attacks.
More explanations to fry your brain, if it is not there yet, here:
http://www.earlham.edu/~peters/courses/logsys/g-proof.htm

olanv
01-16-2004, 05:48 PM
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?

micilin
01-16-2004, 09:27 PM
justhink... is that you?

Pedro
01-17-2004, 12:39 AM
In other words, you declare that within the system, the axioms have a truth value of 1, and that any statement that can be generated through proper use of the rules also has a truth value of 1. Using the axioms and rules, as proved rigorously by Goedel, you can, for some statement x, create the statements "It is true that x" and "It is not true that x", both with truth value 1. We must conclude from this contradiction that the truth value of x is undecidable.

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).

Spiritus Mundi
01-17-2004, 02:41 AM
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:
P
P -> Q
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.

It doesn't matter "only in relation to k", because what's being argued, is that the assent of undeterminability is the assent to something that circumvents the entire axiomatic process and the inference process. It would make inference a zero set, which you already said it cannot be.

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?

Undecidability is necessarily referencing the concept of undeterminability.

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.

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.

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.

[That undeterminability EXISTS] means that an aspect of existence is not bound by any possible inference or causality or axiom.

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.

Which means that the claim of uncertainty is itself uncertain.

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.

scotandrsn
01-17-2004, 10:57 AM
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.

scotandrsn
01-17-2004, 11:43 AM
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.

scotandrsn
01-17-2004, 02:05 PM
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.

olanv
01-17-2004, 11:56 PM
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.

olanv
01-18-2004, 12:19 AM
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.".

ultrafilter
01-18-2004, 01:11 AM
But, think about this. If the rules collapse in one system, then it begs the question of whether the rules collapse in all systems.

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....

Spiritus Mundi
01-18-2004, 02:46 AM
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.

Spiritus Mundi
01-18-2004, 02:54 AM
I should ammend: [i]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.

olanv
01-18-2004, 10:49 AM
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....

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

olanv
01-18-2004, 11:00 AM
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.

ultrafilter
01-18-2004, 03:34 PM
What does it MEAN to say that all symbols are equivicated?

I don't know, because I have never said that.

olanv
01-18-2004, 04:28 PM
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.

olanv
01-18-2004, 04:31 PM
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!!

ultrafilter
01-18-2004, 04:48 PM
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.

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?

Cabbage
01-18-2004, 05:02 PM
Let me try my hand in this. olanv's argument (as I understand it, anyway) hasn't been directly addressed in my (admittedly brief) reading of this thread, so I'll try to address it specifically.

olanv, you seem to have a fundamental misunderstanding of what Goedel's Incompleteness Theorem (GIT) says. Your argument, paraphrased, seems to be the following:

1. GIT claims that all propositions in any ("strong" enough) formal system are undecidable--we are unable to determine the truth value of any statement.

2. Since GIT itself is one of those statements, we, in fact, cannot even determine that GIT itself is true.

3. Therefore, GIT collapses under its own undecidability.

In fact, your argument (assuming this is your intended argument), is perfectly valid.

The flaw, however, is in your first premise. You have misunderstood/misrepresented GIT.

GIT does not claim that all propositions are undecidable, only that there are some undecidable propositions. In fact, using GIT, we can conclude that any proposition falls into one of three categories:

A. Those propositions that are provably true within the formal system.

B. Those that are provably false within the formal system, and

C. Those that can neither be proven true nor proven false within the system (the undecidable propositions that GIT guarantees).

GIT only claims that there are certain propositions that are undecidable, not that, as a general rule, all propositions are undecidable. There still remain propositions that we can prove are true (such as GIT itself), as well as propositions we can prove are false.

How do we know which of the three categories a given proposition may fall in to? In general, it's not at all easy.

In certain cases, however, it's really quite simple. If we can construct a proof of the truth of a proposition, that's all we need to establish that that proposition belongs in category A. GIT itself falls into this category, simply because of the fact that it has been proven. Similarly, if we can prove that a given proposition is false, we have demonstrated that it lives in category B.

Unfortunately, in general, if we're given some arbitrary proposition, there's no systematic way of demonstrating what type it is (i.e., which of the above categories it falls into). It can often be done (it's been done many, many times, as evidenced simply by the sheer number of theorems throughout mathematics), but, in general, it can't be done.

Demonstrating that a proposition falls into category C is often the most difficult to establish. A standard approach is the following. Given a proposition P:

1. Construct a model M which does two things: a. Satisfies the axioms of the formal system, and b. Satisfies P (in other words, P is true in this model).

2. Also, construct a model M' which does two things: a. Satisfies the axioms of the formal system, and b. Satisfies (not P).

1. demonstrates that P is consistent with the formal system; 2. demonstrates that (not P) is consistent with the formal system. Together, 1. and 2. demonstrate that P is undecidable in the formal system--one way to think of it, in some sense, is that the formal system is not "strong" enough to distinguish between, for example, the models M and M', and is therefore not strong enough to determine the truth value of P.

In fact, to give a specific example, this is exactly the method used by Goedel and Cohen to prove the undecidability of the Continuum Hypothesis (CH) in Zermelo-Fraenkel-Choice (ZFC) set theory. In the 30's (I believe), Goedel showed that CH was consistent with ZFC; in the 60's, Cohen showed that (not CH) was consistent with ZFC. Together, these two results demonstrate the undecidability of CH in ZFC.

olanv
01-18-2004, 07:19 PM
1. GIT claims that all propositions in any ("strong" enough) formal system are undecidable--we are unable to determine the truth value of any statement.

......

GIT does not claim that all propositions are undecidable, only that there are some undecidable propositions. In fact, using GIT, we can conclude that any proposition falls into one of three categories:

A. Those propositions that are provably true within the formal system.

B. Those that are provably false within the formal system, and

C. Those that can neither be proven true nor proven false within the system (the undecidable propositions that GIT guarantees).

GIT only claims that there are certain propositions that are undecidable, not that, as a general rule, all propositions are undecidable. There still remain propositions that we can prove are true (such as GIT itself), as well as propositions we can prove are false.


You're absolutely correct! That's my claim. However, my claim is also that it's impossible to conclude undecidability without referencing the concept of undeterminability. Undecidability is a decidedly anthropomorphic sense of undeterminability. The begged question is, does undeterminability even exist as a concept that doesn't always and instantly refute itself? Which also directly impacts the question of undecidability (what does undecidability then mean if not defined as "an accepted measure of undeterminability within the scope of thought, reason or logic?")

This is my point. If you even suggest that undeterminability exists in some way that is not self refuting... then you have effectively opened up a logical vaccuum in the universe of logic that can arbitrarily dump rules that effect not only strong systems but even the weakest ones in a manner that refutes your claim of the existence of undeterminability. What this ultimately suggests is that undecidability as a concept is bunk. Someone's making a mistake, somewhere.

What I'm saying, is that when playing 'games' with a concept like undeterminability, there is no "acceptable" measure of allowance for this concept in the assertation of truth.


Do I know exactly where someone screwed up? No. But through this reasoning, I can conclude that someone must have screwed up their logic somewhere... because per my observation, undecidability implies undeterminability and undeterminability is self refuting.

Cabbage
01-18-2004, 11:25 PM
Well, I had thought that maybe (just maybe) I was following your argument, however, your last post has shattered that thought. I do not follow your "explanations" at all. This statement, in particular:If you even suggest that undeterminability exists in some way that is not self refuting... then you have effectively opened up a logical vaccuum in the universe of logic that can arbitrarily dump rules that effect not only strong systems but even the weakest ones in a manner that refutes your claim of the existence of undeterminability. What this ultimately suggests is that undecidability as a concept is bunk. Someone's making a mistake, somewhere.
makes no sense to me whatsoever. What, exactly, are you attempting to say?

Several posts ago, you asked others to, "Define: Undeterminability, Non-Determinability, Adeterminability, Anti-determinability, Indeterminability", however, it's you that have been tossing those words around (along with others, such as undecidability) as if each of them has some distinct, distinguishing meaning. What do you mean by them?

Anyway, and perhaps more to the point, here's a fairly simple (and informal) example. Say we have the following axioms:

1. All boojums are snarks.
2. Lewis is a boojum.
3. Carroll is not a snark.
4. Dodgson is a snark.

Now consider the following propositions:

A. Lewis is a snark.
B. Carroll is a boojum.
C. Dodgson is a boojum.

We wish to determine the truth value of each of the propositions A, B, and C.

I can prove A:

Lewis is a boojum. (Axiom 2)
All boojums are snarks. (Axiom 1)

Therefore, Lewis is a snark.

I can disprove B:

Suppose Carroll is a boojum.
All boojums are snarks. (Axiom 1)
Therefore, Carroll is a snark.
However, Carroll is not a snark. (Axiom 3)
Contradiction, hence our original supposition is false, and therefore Carroll is not a boojum.

So A is True, B is False. What about C?

I'll leave C as an exercise for you, olanv. Surely, according to you, C must be True or False within our system (since Undecidability doesn't seem to be an option according to you).

Which is it? True or False? Prove your answer.

Spiritus Mundi
01-19-2004, 12:20 AM
The begged question is, does undeterminability even exist as a concept that doesn't always and instantly refute itself?

The question is not begged. If you are going to hurl stones at the towers of logic you migt want to at least acquaint yourself with the terminology of the field. It might mitigate, somewhat, the impression that you are entirely lacking in the erudition necessary to develop your arguments in a rational and compelling manner.

Setting aside issues of word choice, I will go ahead and answer your question. Yes. Undeterminablilty (or indeterminism, if you prefer) does exist as a concept that does not "always and instantly refute itself".

In fact, several of us have given specific examples of systems containing indeterminism. Thus far, you have exhibited an impressive prediliction for avoiding comment upon even the simplest of such systems. But let's give you another chance, eh? Would you please demonstrate how the inability of my trivial little system above to assign a truth value to R "instantly refutes itself".

I shall await the results of this test of intellectual character with great anticipation.