Scott,
I honestly and swear on my mother's grave that I am 1.) Not trying to flame you. 2.) Not trying to insult you. I do not know any other way to say this to you in a way that is "nice". So I will just say it, and you can accept my words or not.
Your reply is a mish-mash of confusion between logic and and these things called deductive systems. You keep saying things like "When Godel extended logic" and you refer to "Godel's system". These phrases
make no sense. I have little indication that you have understood anything I have said to you in the previous post. I could go line-by-line but I don't see the point, because it was just be me repeating this same message over and over again.
I have already mentioned several issues of LOGIC which exceed the scope of Godel's theorem. The first example was an agent who acts in a world under uncertainty, and whose beliefs changes whenever new data comes in. That agent's logic is going to be very different from a static system that is assumed to contain a priori all of its consequences hanging motionless in a platonic superchest. (if you will)
The underlined "uncertainty" implies incompleteness and potential inconsistency.
It absolutely implies no such thing. But again, you are not even using these phrases correctly. You totally overlooked that I had also underlined (and capitalized)
issues of LOGIC. Because you are constantly posting erronious things about the application of Godel's theorems to logic. I understand clearly why these two things are different, but your posts indicate a confusion.
I thought you understood I was referencing ALL reality, not simply math. ?? Yes, the theorems hold for the physical laws of science too. As long as they are consistent logics, Godel's system holds true for all such logic.
I am not the confused person here.
Listen. Bertrand Russell and Alfred North Whitehead discovered that all branches of mathematics, which were once considered separate disciplines, were all united under the same umbrella. That umbrella is something called Zermelo-Fraenkel-Choice Set Theory. By the time of Hilbert and Godel, proof theory had developed far enough that mathematicians could "step outside" of ZFC and refer to is as a "system". A "system" contains symbols, sentences, well-formed-formulas, and deductive translations between them. (To be more formal and academic with you, because that's important right now) we should laboriously refer to ZFC as a deductive system.
Completeness
If a deductive system is "complete", then all the sentences within the system which are true can be derived from its axioms by deduction. (If s is a logical consequence of the system, then there exists a proof of that fact).
Consistency
If a deductive system is "consistent", then for all sentences s, which contain a proof showing that s is true, there will not exist a proof showing that s is false.
Theorem
If a sentence s is in a deductive system U has a proof, then s is a theorem in U.
Incompleteness
If U is incomplete, there will exist an s in U that is a true statement which is not a theorem.
Undecidable
A sentence s is "undedicable" if and only if "s is true" is not a theorem in U, and "s if false" is not a theorem in U. In most working cases, both of these must be proven. In some cases in history, both proofs are proven decades apart by different people.
Independent
A sentence s is "independent of U" if and only if s is undecidable in U
and s is consistent with the axioms of U.
The above glossary should go a long way. I have warned you earlier in this thread, and I will warn you a second time. Wikipedia is absolutely terrible for this topic -- they write about all this stuff in an "ultra-modern" form, which assumes the reader has had exposure to Model Theory. (I do not doubt that Godel's work as a mathematicians has , in the ensuring decades, been formalized as 'obvious things' in Model Theory. ) But generally, the wikipedians are being obtuse!
In your reading of the material surrounding this topic, you will undoubtedly come upon something like :
"Godel showed that the Continuum Hypothesis is independent of ZFC."
Armed with the above glossary, you should be able to parse this sentence now. Let CH=Continuum Hypothesis. You should see that CH exists as a well-formed sentence of ZFC. Yet ZFC cannot be used to prove CH is true. Further, CH does not refute ZFC.