How would logic represent this idea?

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.

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.

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.

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.

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.

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.

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.

Not quite–I don’t have time to go into it right now, but I will later.

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.

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.

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.

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.

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!!

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.

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.

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.

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).

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.