# Thread: Do IEs Really Exist?

1. Originally Posted by Singu
The Incompleteness Theorem says that you can never prove any consistent system to be either true or false, they are "undecidable".
No it doesn't say that. It's false over-interpretation of Godel first theorem. Godel never said that nor mentioned in his proofs.

2. Originally Posted by Singu
...It has everything to do with Godel's Incompleteness Theorem. The Incompleteness Theorem says that you can never prove any consistent system to be either true or false, they are "undecidable". So the only way to "prove" (of course there is no real such thing) something is to actually test it out in reality and see how it would respond. It also says that for every proofs out there, there exists an equal amount of proofs that would invalidate those proofs.
Once again you have no idea what you're talking about. The theorem says that a (sufficiently expressive and consistent) system can't prove itself to be consistent. One system can prove another one to be consistent but it will essentially be stronger than the one it is claiming the consistency of. Truth is defined only when a particular interpretation of a theory is given (i.e. comparing it against reality).

3. Socionics is an attempt to apply logic (rules) to the +Ni analogic stream that composes the undercurrent of the world. The problem is that a system of rules can't never measure reality 100%, since there are too many unseen factors. But we can't operate the undercurrent directly; we need structures and models that help us interpret reality, and allow us to make predictions. But these have limitations. All the physics theorems that we have and seem to work, do so in an standalone manner and some are incompatible with others (for example; quantum mechanics and general relativity). It's like when we made the switch to digital video from analogic video; analogic video contains much more "info" than digital (even if digital seems more sharper at first glance). Same with digital audio; information is "lost" in the encoding. That is why when you go to theatre to listen to a concert, you feel much more emotion than if you listen to an mp3 of the same act.

4. Originally Posted by thehotelambush
Once again you have no idea what you're talking about. The theorem says that a (sufficiently expressive and consistent) system can't prove itself to be consistent.
That's because you didn't understand the Godel's Incompleteness Theorem properly, or you misunderstood how it works.

This is what the Theorem says:

1. In any sufficiently complex system, a system is either incomplete or inconsistent.
2. Such a system can't prove its own consistency ("Consistent" in this sense means that you can not prove both a statement and its opposite to be true, as it would be contradictory e.g. you can't prove that both unicorns exist and unicorns don't exist).

You're talking about #2. But #2 is just an extension or the consequence of the #1, and it's not the main point.

--

What Godel did was through a very complicated process, he has devised a way to prove any statements to be either true or false. Let's call it the function Provable(x), that can prove any statement to be either true or false. He has also devised a way to convert any statements into numbers, just as computer codes can be converted into 1's and 0's these days (this is called the Godel-numbering), so that we can simply plug in the numbers to the Provable(x) function. But instead of using numbers, we'll just use English letters for now because it's easier for us to understand. But in principle, it's possible to use numbers and thereby making it a mathematical statement.

He has also created the opposite of Provable(x), the NOT Provable(x). Let's call it the function NotProvable(x). Since this is just the opposite of Provable(x), it just reverses the answer. If NotProvable(x) = true, then Provable(x) = false, and vice versa.

This is all very good and all, but what can we do? Godel has done all this just so that he can prove a very important point.

Let's just suppose that NotProvable(x) is false. We want to say that NotProvable(x) was proven to be false by plugging it in to the Provable(x) function. What will happen then?

If NotProvable(x) = false

NotProvable(x) = false (= Provable(x) = true)

equals

Provable(NotProvable(x)) = true

equals

NotProvable(x) = true
(To get a better picture, NotProvable(x) is like saying in plain English, "This statement is false". If we say that it's false, then it's true.)

So this results in NotProvable(x) being true, even though we said it was false? This is a logical contradiction and an inconsistency, since we have just proved that something that was supposed to be false, is true. That means we can prove a false statement to be true. This will be the end of math as we know it.

We don't want that to happen, so let's assume that NotProvable(x) is true instead. What will happen?

If NotProvable(x) = true

NotProvable(x) = true

equals

Provable(x) = false
This is a better option, but it also means that NotProvable(x) can't be proved to be either true or false, since we can no longer use the Provable(x) function. It will forever remain a mystery.

We are only supposing that NotProvable(x) is true, only because we don't want an inconsistent system. There is no "proof" of this anywhere, in fact we have brought this "NotProvable(x) = false" from outside of the sytem. It is true, but unprovable. Therefore, the system is incomplete. There is a statement that can't be proven within the sytem.

And that is why a system is either incomplete or inconsistent. It also can't prove its own consistency, because if we did prove its own consistency, then as in the "NotProvable(x) = false" example, it will prove to be inconsistent. But we'd want a math system to be consistent, so we are only supposing that it's consistent, but also remaining incomplete forever (we can no longer prove it). Since it's not provable, we'd have to keep adding new axioms from outside of the system forever.

Originally Posted by thehotelambush
Truth is defined only when a particular interpretation of a theory is given (i.e. comparing it against reality).
In a theory, there is no such thing as a "truth". Perhaps we can say that either it's "good enough" for something, or it's an approximation of the truth. Strictly speaking, a theory is always "wrong", and there could be another theory that is "less wrong" than the other one. But as the Godel's Incompleteness Theorem says, we can't ever reach the "truth", as we keep have to incorporating new information from outside of the system, ad infinitum. It will always remain "incomplete".

In math and logic, a truth value is defined by its axioms.

5. Originally Posted by Bertrand
the theorem is more about how the more you zoom in and gain accuracy on that front you lose resolution on context
That's just stupid on a whole new level.

6. Originally Posted by Singu
That's just stupid on a whole new level.
you're going to realize like in a few decades (optimistically) my statement was correct and feel bad, but don't feel bad, know I'm smiling somewhere

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