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Godel's Theorem for Dummies.


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#1 Vexer

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Posted 22 April 2008 - 08:28 AM

It says that a 'system' cannot be understood (or 'described') without the 'rules' of a 'higher' system.


1. Give me a better, but equally short definition.

2. Well.... I guess '2' depends upon 1.


I want to know.
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#2 iNow

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Posted 22 April 2008 - 01:45 PM

Any object being described is, by definition, a subset of the system in which the description is being offered.
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#3 bascule

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Posted 22 April 2008 - 05:20 PM

Godel found a way of encoding a statement to the effect of "This statement is unprovable" into the symbolic logic system defined in Principia Mathematica (PM). The notable aspect of the statement is that it is self-referential, which Godel managed to accomplish by encoding statements in PM into "Godel Numbers." Thus the actual statement in PM refers to its own Godel Number.

To boil it down into a nutshell, I'd say it means that any system which is expressive enough to be consistent and complete is also expressive enough to contain self-referential statements which doom it to incompleteness.
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#4 Vexer

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Posted 25 April 2008 - 09:33 AM

So everything is unknowable. In effect.
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#5 bascule

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Posted 25 April 2008 - 07:49 PM

So everything is unknowable. In effect.


Not really "unknowable". The point is that no formal logic system can be both consistent and complete. Godel proved that any system with the expressive power of basic (Peano) arithmetic is also expressive enough to contain unprovable statements.
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#6 Mr Skeptic

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Posted 28 April 2008 - 02:04 AM

So everything is unknowable. In effect.


No, for example I know that statement is false (and how could you say that since you wouldn't know it was true?). What it says is that, a logic language cannot be expressive enough to contain self-referential statements, be consistent (not have contradictions), and be complete (everything is provable/knowable).

Basically, that something is unknowable, in said logic language.
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#7 Vexer

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Posted 28 April 2008 - 10:22 AM

I know you’re going to be annoyed:

The point is that no formal logic system can be both consistent and complete. Godel proved that any system with the expressive power of basic (Peano) arithmetic is also expressive enough to contain unprovable statements.

That’s what I said. Essentially, by definition, everything is essentially unknowable.


Still seems ok by original statement, Mr Skeptic.
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#8 Mr Skeptic

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Posted 28 April 2008 - 03:16 PM

Sure, if you can't tell the difference between everything and something. And it has nothing to do with definition of anything except the Godel Statement.
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Our voting system is broken! It nearly guarantees that we will have only two political parties that have any chance of winning, and that they will be very similar.

#9 bascule

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Posted 28 April 2008 - 08:32 PM

The point is that no formal logic system can be both consistent and complete. Godel proved that any system with the expressive power of basic (Peano) arithmetic is also expressive enough to contain unprovable statements.

That’s what I said. Essentially, by definition, everything is essentially unknowable.


I think you're missing the distinction between unknowable and unprovable
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#10 Vexer

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Posted 2 May 2008 - 09:06 AM

I think you're missing the distinction between unknowable and unprovable

The same thing.
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#11 Vexer

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Posted 10 May 2008 - 07:55 AM

bascule, you're initial response is too good. (I'm going to steal it). Brilliant. (I mention this now, because I didn't mention it before. This is the kind of response I hope for).
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#12 Aeternus

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Posted 11 May 2008 - 11:21 AM

Shouldn't it be mentioned that the theorem initially mentioned in the first post refers to Gödel's Second Incompleteness Theorem (which uses the first but isn't the same theorem), while later posts refer to the First Incompleteness Theorem.

Also "Gödel's Theorem for Dummies" is a little non-specific as he is known for several other theorems including his Completeness Theorem.
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#13 Vexer

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Posted 13 May 2008 - 08:07 AM

A Dummy wouldn't know that.

I'm interested in the Big Picture, and this seems relevant.
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#14 yrreg

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Posted 2 July 2013 - 05:03 AM

I think you're missing the distinction between unknowable and unprovable

The same thing.

 

 

 

To know something in the broadest sense of the word means to have any idea at all about that something.

 

For example, atheists call God a flying spaghetti monster, they know what is a flying spaghetti monster, it is all in their mind, but they have not shown a flying spaghetti monster yet as to prove to mankind that God is such a flying spaghetti monster which they have located in the reality of existence outside their mind.

 

So, to know something is to have any idea at all in the mind of humans and also in the reality of existence outside the mind of humans.

 

Now, to prove something is to show that the concept of something in the mind (we'll talk about in reality later) is not internally inconsistent and incoherent namely, that it is internally consistent and coherent.

 

For example, the popular example of an internally inconsistent and incoherent concept is the square-circle, it is a circle and a square at the same time and in the same aspect of consideration: such a figure cannot be a square if it is a circle, and not a circle if it is a square.

 

But we still in a way know what is a square-circle, not in itself as something that is internally inconsistent and incoherent, but by comparison with internally consistent and coherent things.

 

If you talk about a square-circle, no one can know what you are talking about, except that you must be into showing them what is an internally inconsistent and incoherent concept compared to internally consistent and coherent concepts.

 

We now come to what it is to prove something to exist.

 

Ask ourselves the question first, exist where? In the mind or in the reality of existence outside our mind, or both in the mind and in the existence of reality outside our mind?

 

1a. How to prove something exists in our mind: show to your listeners that they can also think of such a thing in their minds, it is thinkable and they and you can actually think of it and talk about it, the concept in our minds.

 

For example, a human with the abdomen upward human but with the part below a goat (satyr): we can keep such an idea and even image in our mind and talk about it.

 

1b. How to prove something to exist in our mind: by showing that it is implied or even identical to another thing which we know about; for example, you know a grown-up man, that implies that you know him to be once upon a time a male infant; then also you can show that the man and the male infant are identical.

 

2a. How to prove that something in the mind is present in the reality of existence outside our mind, for example, Bigfoot or Susquatch: you and I go forward in the reality of existence and look for him, because he is described as bigger than ordinary humans and has big feet, still essentially human -- if we don't find him, then we have to search farther and more intensively and more continuously, but for the time being we can say that we have not found him: we cannot say that he does not exist, because Bigfoot or Susquatch is in concept absolutely internally consistent and coherent, he is not any internally inconsistent and incoherent in concept.

 

2b. What about things which we cannot locate in the reality of existence outside our mind but they are supposed to be existing, like for example, God, or in olden days, bacteria as yet then unknown to mankind?

 

First, something can be existing but we cannot experience it directly with our senses, because it is too big or too small.

 

In the case of God, He is at the same time too big for man to be able to notice His presence, and at the same time He is so subtle as to be also for man impossible to notice His presence. How big is God? bigger than the universe as to contain the universe; and how small? smaller than whatever space prevails between the smallest sub-atomic particles still not yet discovered.

 

You notice that with bacteria, no one could prove directly their existence until the invention of the microscope.

 

So, how do we know enough as to prove the existence of God, and of the bacteria prior to the invention of the microscope? From the effects brought about by God, and also by bacteria in their own case (there are bacteria advantagious to mankind and there are bacteria harmful to mankind.

 

Now with God He is so big and so small and everywhere that we cannot notice His presence, but His effects are seen in the universe which He created and everything in the universe, including our nose, so that as you see your nose you can tell God, "Thank you, God, for the nose in my face."
:

 

 

So, what do you guys here say about my thinking on what is knowable and what is provable.

 

 

 

 

Gerry


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