Lib
I think I can connect the dots. Let’s see what the picture looks like.
As you yourself noted, that sequel never appeared. Extending GIT to generalized epistemologies is not a trivial exercise. If Godel ever managed to do so, he left no record. Many people have “rephrased” GIT to apply to whatever general sphere they wish to call udecidable, but without the logical demonstration such extensions are not valid.
It seems like GIT should extend generally to complex epistemologies, but I do not believe it has ever been demonstrated.
That is not the correct question, though. The question is must a finite epistemology contain paradox? The answer to that question is no. It is not difficult to construct a finite epistemology which contains no paradox. It is difficult to construct one which contains no paradox and is complex enough to be useful. It might be impossible to do so, as you assert, but I know of nobody who has proven that.
I have been saying what I mean. Do you object to my asking you questions which are intended to illuminate an area in which we disagree?
“Whatever else” the brain does is not irrelevant when speaking of proving the consistency of a set. “Whatever else” is exactly how we demonstrate that any particular set is consistent. As I noted, this process of supersetting to demonstrate results has no finite end, and thus no end accessibly to the human mind. The existence of an absolute external authority does not provide an escape for this, since the human understanding of said absolute remains bound by the epistemological shell.
Actually, Peano Algebra is demonstrably consistent under ZFC. That is, in fact, why Godel’s theorems are called incompleteness theorems rather than inconsistency theorems.
The relevance is that PA provides an example of how the human brain uses something outside of PA to demonstrate the consistency of PA. No god required. It is also an illustration that “undecidably consistent from within” carries no implication of “inconsistent”.
Interestingly, one element of ZFC which allows us to demonstrate the consistency of PA is the ability to speak about infinite sets.
GIT is second only to quantum dynamics as a source for loose “summary”, IMO.
Siegfried carefully includes “sufficiently complex” and “consistent” in his formulation. Sufficiently complex, in discussion of GIT, is almost always shorthand for “capable of forming a Godel statement.” Now, you have already denied your ability to frame a generalized Godel statement over the set of all moralities (or epistemologies). To the best of my knowledge, nobody else has formulated such a statement.
This is not a minor quibble. Without such a formulation, you abuse GIT by applying it generally to all epistemologies/moralities.
Yes. But I also understand, and have tried to emphasize, the distinction between undecidably consistent from within and consistent. More than that, using GIT[sub]2[/sub] to argue against a finite moral set on the grounds of inconsistency is absurd. GIT holds only when the set is consistent. It can be used to show that a set is inconsistent only if a proof of consistency for the set appears within the set.
In my ironic moments, I like to contemplate the consequences of GIT[sub]2[/sub] on the infallibility of books which claim to be infallible.
Hmmm, doesn’t “reliable” imply that someone is relying upon it? I thougt we were discussing the limits of human knowledge. Neither of us has anything rigorously meaningful to say about the nature of God’s knowledge.
Well, if nothing is revised then your statement remains invalid.
Nothing in GIT lets you conslude that no moral distinction can be drawn from a finite moral set.
And I’m not even going to begin the process of examining how a human being declares a moral set to be immoral. It’s turtles all the way down.
{X:“reading this sentence is the only good”}
Not particularly useful, but finite and devoid of antinomy.
More to the point, though, even if your statement was true it does not carry the consequence of paradox to any arbitrary finite moral set.