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Unfolding Nature Shop: Unfolding Nature: Being in the Implicate Order

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Godels Theorem
    #14996402 -

Anyone familiar? I thought I had a pretty good grasp of it but im not so sure right now. heres a wiki:
Quote:
Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete. According to the theorem, for every sufficiently powerful logical system (such as Principia), there exists a statement G that essentially reads, "The statement G cannot be proved." Such a statement is a sort of Catch-22: if G is provable, then it is false, and the system is therefore inconsistent; and if G is not provable, then it is true, and the system is therefore incomplete.

Gödel's second incompleteness theorem (1931) shows that no formal system extending basic arithmetic can be used to prove its own consistency. Thus, the statement "there are no contradictions in the Principia system" cannot be proven in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).


It seems like a truism that A statement cannot prove itself. Proof involves associating other statements together. Self-reference buggers up a logical system, eg. when we try to talk about our qualia. So we need to just be a bit inconsistent, ie, use a slightly incorrect word in order to get as close as we can to being self referential.

As for the second incompleteness theorem, once again, we have a self referential statement that just wont compute.

All other statements are referring to other statements, thus building relationship networks with eachother. But when a statement is isolated from the rest, by referring to itself, it ceases complete computation.


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Re: Godels Theorem [Re: Noteworthy]
    #14996868 -

It seems like a truism that A statement cannot prove itself. Proof involves associating other statements together

I've written several threads on this topic here throughout the years. Search my posts and you'll find a lot of stuff to read. But for now, consider the following statement:

If x is true, then x is true

Do you accept the statement is valid? If so, notice that you didn't need any other statement to reach that conclusion. One of the foundations of modern mathematics is called First-order Predicate Logic. I'm tempted to go off on one of my long-winded explanations, but they take me a very long time to put together and proof-read, and I'm only willing to do it when I'm stoned, which I'm not at the moment. :bigweed:

But if you read that Wiki entry, you'll grok the extremely deep philosophical consequences of Godel's proof. And I'll answer any questions if I know the answer.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #14997179 -

I love this stuff and can't resist writing some more. For the non-geek, Godel's proof can be summed as follows:

Imagine you're the head librarian of a city with many libraries. You've asked each branch librarian to compile a catalog of all the books in their particular library.

When you receive them all, you find that some librarians listed their catalog in itself, since the catalog is itself a book after all. Others decided not to list the catalog in itself.

So, now you have two stacks of catalogs, one where they list themselves, and another where they don't.

--

Your task is to compile a master catalog of only the catalogs that DO NOT list themselves; a meta-catalog of sorts. A catalog of catalogs.

When you finish the meta-catalog of all the catalogs that DO NOT list themselves, you realize that your meta-catalog is incomplete because it's missing itself. Remember that you're compiling a meta-catalog of catalogs that do not list themselves. But the instant you add the meta-catalog to its own list, it becomes incorrect because again remember it's a meta-catalog of catalogs that DON'T list themselves and now that you listed it in itself, you rendered it incorrect.

So you have a quandary: either the catalog is incomplete (it's missing itself) or it's incorrect (it contains itself).

This self-referential construct is a very deep concept in mathematics and it's at the heart of Godel's theorems. It turns out that no system of mathematical rules can ever be complete because adding the final rule makes the system incorrect, just like the catalog.

Weird eh? If there is a Creator, he definitely has a sense of humor.

Now where's my bong? I feel like writing some more. :hehehe:


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Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #14997879 -

"But if you read that Wiki entry, you'll grok the extremely deep philosophical consequences of Godel's proof. "

As much as I love to apply Godel to everything, ask anyone who has studied the proof and they will assure you it applies only to mathematics. :shrug:

Also I was a bit confused by your analogy, let me share one I've seen.

A barber cuts everyone's hair who does not cut their own hair. Does the barber cut his hair?

I've been told thinking about that analogy pretty much shows the godelean idea of incompleteness. If the barber cuts his own hair, then he would not cut his own hair. If he does not cut his own hair, he could cut his own hair. This is supposed to be like the set of all sets that don't contain themselves.

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Re: Godels Theorem [Re: xFrockx]
    #14997921 -

Quote:
ask anyone who has studied the proof and they will assure you it applies only to mathematics.


As much as that is true, it in no way mitigates the deep philosophical consequences.

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Re: Godels Theorem [Re: Diploid]
    #14997998 -

Diploid said:
For the non-geek, Godel's proof can be summed as follows:


So its a paradox of sorts?

Let me try: God created everything, but God wasn't created.
A)God didn't create everything, because he didn't create himself (incomplete)
B)God did create everything, including himself (wrong, because he wasn't created)

Right track?


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"Sacrifice is the Christian way of life.  Theft is the Socialist way of achieving all ends.  Together they represent an awesome evil for this nation to confront."  Marc Emery.

People who receive medical cannabis trough the federal government.

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Re: Godels Theorem [Re: Diploid]
    #14998118 -

I've been trying to fully understand Gödel's Theorem for a long time, and as for someone who's interested in Philosophy and is pretty poor at math past basic algebra, you explained it clearly. Thanks a lot! :awesomenod:


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Current favorite candy: Peanut Butter Kisses

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Re: Godels Theorem [Re: r72rock]
    #14998135 -

You and I are in the same boat.


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"Sacrifice is the Christian way of life.  Theft is the Socialist way of achieving all ends.  Together they represent an awesome evil for this nation to confront."  Marc Emery.

People who receive medical cannabis trough the federal government.

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Re: Godels Theorem [Re: Diploid]
    #14998909 -

The analogy of catalogs that contain themself reminds me of Russells paradox. How is what you're talking about different?


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Re: Godels Theorem [Re: Diploid]
    #14998997 -

Diploid said:
It seems like a truism that A statement cannot prove itself. Proof involves associating other statements together

I've written several threads on this topic here throughout the years. Search my posts and you'll find a lot of stuff to read. But for now, consider the following statement:

If x is true, then x is true

Do you accept the statement is valid? If so, notice that you didn't need any other statement to reach that conclusion. One of the foundations of modern mathematics is called First-order Predicate Logic. I'm tempted to go off on one of my long-winded explanations, but they take me a very long time to put together and proof-read, and I'm only willing to do it when I'm stoned, which I'm not at the moment. :bigweed:

But if you read that Wiki entry, you'll grok the extremely deep philosophical consequences of Godel's proof. And I'll answer any questions if I know the answer.


Since you are interested, do you think you could expound on the consequences of Godels proof?

What positive results have come from Godels theorem, other than claims about the nature of predicate logic?


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Re: Godels Theorem [Re: Noteworthy]
    #14999481 -

Now you're definitely asking for a stoneded rant and that will have to wait until my bong is reloaded and I have the time to write. The problem is that while the theorem can be superficially understood without the language of mathematics, to get at the deeper aspects you need the technical background to make the discussion meaningful. If you don't know what I mean when I say Zermelo-Fraenkel Set Theory with Choice, Peano Arithmetic, or the Continuum Hypothesis then a deep discussion is out of the question.

But I'll give you some history that might put the theorem's importance to philosophy into context and then a concrete example. More than that is the subject of a textbook sized post so instead I'll refer you to your local community college or university. That, plus I'm far from an expert so maybe someone else can fill in the rest.

So here first some history.

Back in the early 20th century, two brilliant philosophers, Alfred North Whitehead and Bertrand Russel decided to, once and for all, elucidate THE foundation of mathematics. To that end, they started work on their opus magnum. It was called Principia Methematica, not to be confused with Isaac Newton's work with a similar title. This is perhaps one of the most important works in the history of philosophy.

Anyway, back then, the state of affairs in mathematics was one of contradiction. This is anathema to logic. You can't have a logical argument that shows how 1 + 1 = 2 and another logical argument that show how 1 + 1 = 5 unless one (or both) are incorrect. It was a big problem because if there are inconsistencies, then how can you rely on any mathematical result? There's more history leading to this, but it's too much to type so look up Gottfried Leibniz, Gottlob Frege if you're interested.

To resolve the problem, Whitehead and Russel started their work with the goal of removing all inconsistencies from all of mathematics. They toiled for some 20 years producing three monumental volumes in logic that literally looked like pages of wallpaper. If you've ever seen logic notation, you'd know what I mean (see image below). To give you some idea of the scale and detail of this work, consider that in volume 1, it took over 300 pages to PRECISELY define addition and prove conclusively that 1 + 1 = 2. Here are the last few lines of the proof. Notice the last line in particular:



So after all that, along came a young whipper-snapper named Kurt Godel who royally fucked everything up with his proofs (there are actually two of them) which he wrote in his early 20s right after finishing his PhD. It showed that, no matter how big and exhaustive, the Principia will always be unfinished (or wrong). The result shook the mathematical world to its core. It was a deeply disturbing result and Russel all but had a nervous breakdown afterward. Neither he nor Witehead were ever the same again. Hopefully that gives you some idea of the magnitude of this result.

As for a concrete example, consider judicial laws. Laws are written very loosely from a mathematical point of view. In math, everything is defined extremely carefully so that there is ABSOLUTELY NO AMBIGUITY OF ANY KIND. This is hard to do, and it's why human laws written in human language leave so much room for interpretation. But, imagine if laws were written with mathematical precision. For example, the definition of theft would have to start with a definition for the notion of "value" because theft is the taking of something of value. Next you'd have to define what value means. Is it the cost of something? What if you made it yourself out of scraps that cost nothing. Does the work of making it have value? Does the value of that work change depending on the season or the location where it's performed. You see that just the definition of the word "theft" might take a million pages to define with ABSOLUTE PRECISION so that there is nothing at all left to interpretation. Clearly that's not a practical legal system and so we make due with the imprecise but functional law books in actual use.

But imagine if someone took the time to write out our legal system to that level of precision. It might take 100 billion pages to get it all done.

And when you're done, there will STILL be actual legal situations to which two contradicting laws will apply. There will be people before a judge accused of a crime for which one lawyer will present a slam dunk case showing the defendant definitely broke the law, and another lawyer who will use the same laws to show that the defendant did not break the law.

That's one consequence of Godel's work. EVERY POSSIBLE system of "if this, then that" or "if he stole it, then he's guilty" logic will contain similar shortcomings.

Hope that helps. It turned into a long-winded rant after all. No fun sober. I'm going to bed.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Mufungo]
    #14999547 -

The analogy of catalogs that contain themself reminds me of Russells paradox. How is what you're talking about different?

Russell's paradox runs along the same vain but it's not the same paradox. It's properly attributed to Zermelo, but Russel got the credit. It comes about in the naive formulation of set theory and applies strictly to sets. The paradox was later addressed by adjustments to the naive theory, and that was great for a while until Godel came along.

In other words, one is a math problem, the other is a problem with math. That's one of the deepest aspects of Godel's result. It's not a proof in mathematics, it is a proof ABOUT mathematics.

Math studying itself! :whoa:


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #14999567 -

cheers :thumbup:


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Re: Godels Theorem [Re: Diploid]
    #14999568 -

yeah i guess I got into thinking about godels theorem recently because im trying to understand why maths was thought to be based on first order predicate logic. Well i can understand why it was thought to be that way, but I guess im searching for a modern logical basis for arithmetic that avoids the problems posed by principia mathematica and godels subsequent refutation.

Why try to prove 1+1=2 in 300 pages? doesnt that make you think something is wrong? We know 1+1=2 as one of the simplest most elementary logical truths in our whole life.

I commend russell and whitehead for their effort but seriously, if that didnt work, there should be another way about it. So has there been any progress on this front?


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Re: Godels Theorem [Re: Noteworthy]
    #14999611 -

We know 1+1=2 as one of the simplest most elementary logical truths in our whole life.

Well, that's a naive notion.

Consider this. What is 10 + 5? We all know the answer is 15, and it's easy to say that it's (close to) the simplest truth in the world. But that's only in the system of numbers and for the definition of addition we all learned in third grade.

A more nuanced understanding comes from something called modular arithmetic. When you study math, one thing you quickly discover is modular arithmetic in Group Theory. The "Clock Group" is one example.

In geek speak: the Set of positive integers modulus 12 endowed with the operator Addition and obeying the Group axioms of Closure, Associativity, Identity, and Inverses. It works like a 12 hour clock.

What is 10 o'clock + 5? It's not 15 o'clock but rather 3 o'clock. So in the group arithmetic, 10 + 5 = 3.

See how there's a lot of water below the surface? Now I'm REALLY going to sleep. :yawn:


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #14999661 -

actually that seems like an abstract model build on arithmetic plus other notions of identity, recursion, notation, etc.

10 hours past noon on day 1, + 5 hours, is still 15 hours past noon on day 1. It just happens to also be called 3AM (day 2), which is 3 hours past midnight, and 9 hours before noon (-9 hours 'past noon' on day 2)

So although I appreciate that theres more to maths than just arithmetic, your example has not expressed a fundamental difference at the important level. It seems like we build different mathematical models from the same basic arithmetic.

So perhaps another example is needed, whereby arithmetic is trully not being used


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Re: Godels Theorem [Re: Noteworthy]
    #15000288 -

If a system of logic is powerful enough to express statements about itself, then it will have a "blind spot" for dealing with some such statements. Gödel found a brilliant example of such a statement that is unprovable within the system, but obviously true when viewed from another (larger) system.

There are other mathematical statements that cannot be proven from within the system, but which are not obviously true, and have nothing to do with self-referencing. A classic example is the parallel-lines postulate in Euclidean geometry. It can be assumed to be either true or false. If taken to be true, geometry describes a flat surface. If false, a curved surface.

Self-references can produce new degrees of freedom. Consider the self-referential statement S: "The statement S is true." This statement is not obviously true, nor obviously false. Either assumption can be made without causing inconsistencies.

Russell's paradox is related to this. It sneaks in a logically inconsistent statement in a seemingly innocent definition by disguising it behind a universal quantifier and a self-reference.

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Re: Godels Theorem [Re: Rhizoid]
    #15000387 -

so is this all related to a pipe dream of wanting logic to prove logic? or what


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Re: Godels Theorem [Re: Noteworthy]
    #15000757 -

It has nothing to do with wanting or not wanting anything.

It's a statement that relates to fundamental limits to the things that can ever possibly be known, no matter how hard one might to know them.

Limits that are a basic part of of reality. That's why it's so deep.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #15000798 -

I don't know if I'd say the limits are a part of reality, because no matter how reality was we'd still have the problem of Godel. If we tried to explain it all we would be fucked when it came time to explain the explanation, ans explain that, and that... forever.

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Unfolding Nature Shop: Unfolding Nature: Being in the Implicate Order

General Interest > Philosophy, Sociology & Psychology
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