I keep hearing how Gödel is one of the most important mathematicians of all time, rivaling people like Pythagoras and Euler and that in thousands of years time he will be remembered as one of the greatest minds to ever live. But why? For his Incompleteness Theorem?

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Anonymous Sat Oct 12 21:16:29 2013 No.6085996 [Reply] [Original]

I keep hearing how Gödel is one of the most important mathematicians of all time, rivaling people like Pythagoras and Euler and that in thousands of years time he will be remembered as one of the greatest minds to ever live.

But why? For his Incompleteness Theorem?

>>	Anonymous Sat Oct 12 21:17:58 2013 No.6085999 >>6085996 >For his Incompleteness Theorem? yes it was like nothing before it

>>	Anonymous Sat Oct 12 21:23:28 2013 No.6086005 For being the greatest logician that has ever lived.

>>	Anonymous Sat Oct 12 21:25:28 2013 No.6086006 >>6085999 >>6086005 I've not studied his theorem before, why is it so groundbreaking? How come he doesn't seem to get worldwide recognition like other mathematicians? Is this another case of Tesla being under appreciated or something?

>>	Anonymous Sat Oct 12 21:27:28 2013 No.6086008 >>6086006 case like*

>>	Anonymous Sat Oct 12 21:30:29 2013 No.6086011 >>6086006 it said we couldn't prove everything that was true. pretty fundamental

>>	Anonymous Sat Oct 12 21:31:58 2013 No.6086013 What are the implications of his Incompleteness Theorem?

>>	Anonymous Sat Oct 12 21:31:58 2013 No.6086014 >>6086006 Most theorems deal with math. His theorems deal with the mathematical systems we use. Basically he proved that we will either never know whether or not our systems work, or we will know that our systems do not work.

Anonymous Sat Oct 12 22:05:33 2013 No.6086038

>>6086013

Changes the fundamentals of how we view science and math.

Most theorems are based on intricacies, the incompleteness theorem is an observation of the system as a whole. It basically says that although there are axioms (undoubtedly true statements) that we use to prove things, we cannot necessarily prove all our axioms aare true.

His second theorem says that if you use the logic of a system to prove that the system is consistent, then there will be a contradiction somewhere in the system which disproves your logic.

>>	Anonymous Sat Oct 12 22:12:03 2013 No.6086050 >>6086006 >How come he doesn't seem to get worldwide recognition like other mathematicians? He's not unappreciated, Gödel is a household name.

>>	Anonymous Sat Oct 12 22:59:56 2013 No.6086105 >>6086050 >Gödel is a household name. How's life in Bitburg?

>>	Anonymous Sat Oct 12 23:00:57 2013 No.6086107 He created the field of mathematical logic >For his Incompleteness Theorem Of course not

>>	Anonymous Sat Oct 12 23:05:26 2013 No.6086117 >>6086013 There exist theorems that cannot be proven formally but are nevertheless true.

>>	Anonymous Sat Oct 12 23:10:56 2013 No.6086124 Let's get a little more closer to source in this thread... http://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

>>	Anonymous Sat Oct 12 23:36:43 2013 No.6086161 >>6086011 >>6086117 it's a little more subtle than that.

Anonymous Sat Oct 12 23:47:44 2013 No.6086179

Any meta-mathematical proof of PM that can be mirrored inside of PM is inevitably going to be either inconsistent or incomplete, where inconsistent is understood to mean 'containing one or more contradictions' and incomplete to mean 'not containing all truths that are provably true'.

>>	Anonymous Sun Oct 13 00:04:11 2013 No.6086221 >>6086117 is that the nature of logic itself? or will there be newer, better mathematics with "smarter" logic where those "facts" won't be true anymore?

Anonymous Sun Oct 13 00:12:11 2013 No.6086239

>>6086221
Any formal system strong enough to describe arithmetic (eg Peano axioms) will have that problem.

There are weaker theories though ( eg https://en.wikipedia.org/wiki/Presberger_arithmetic, and also Hilbert's axiomitization for Euclidian geometry) that you can recursively enurmerate all possible theorems.

Basically what Godel did was encode "This is a false statement." using the fundamental theorem of arithmetic. Because every natural number factors uniquely, you can use it to encode information of a formal theory.

>>	Anonymous Sun Oct 13 00:13:42 2013 No.6086243 >>6086239 I guess I just don't get it. I'm only a second year maths student. I figure that you could create some logic that transcends the logic those theorems uses and then nothing they'd say would apply.

http://www.youtube.com/watch?v=Bu18cVwbEuU Anonymous Sun Oct 13 00:51:03 2013 No.6086293
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The most beautiful implication for me personally from Godel's work is the following.

Assuming neither nature or math contain contradictions and our brains do, when modeling nature with math + our brains:

>Math can know when it has the right answer, but cannot know all answers.
>Our brains cannot know when we have the right answer, but can know all the answers.

I believe that, someday, this interplay between math, nature, and our brains' physical interpretations and intuitions will be fully appreciated, in the way that Godel's theorem appreciated it, when considering the history of scientific discovery.

>>	Anonymous Sun Oct 13 00:56:33 2013 No.6086299 >>6086038 Sorry, maybe it's because I was educated within the paradigm this shift brought about, but what about this is so groundbreaking? The idea that science and math are based off of unprovable axioms has been around since descartes

>>	Anonymous Sun Oct 13 01:07:23 2013 No.6086317 How do you pronounce his name?

>>	Anonymous Sun Oct 13 01:11:23 2013 No.6086323 >>6086317 It sounds like "girdle."

Anonymous Sun Oct 13 01:12:23 2013 No.6086325

The only people that give a shit about Godel and other logicians are the analytic philosophers from /lit/.

Actual research level mathematicians don't give a flying fuck about him (see comments on sites like MathOverflow).

http://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent/41030#41030

How to spot an undergraduate analytic philosopher on /sci/:

>Goes on about logic whenever people discuss mathematics.
>Name drops Frege or Russell.
>Professes Logicism.
>Pretty much ignores the last 50 years of mathematics, i.e. algebraic geometry, category theory, model theory, proof theory.

Analytic philosophers are the biggest shitposters on both /lit/ and /sci/. On /lit/ they shitpost about science constantly. On /sci/ they shitpost about "muh logic" and "muh philosophy".

>>	Anonymous Sun Oct 13 01:19:35 2013 No.6086335 >>6086325 >anyone who knows anything about analytic philosophy >namedropping Russel you stupid son? captcha: eatsip son

>>	Anonymous Sun Oct 13 02:00:55 2013 No.6086403 >>6086299 Axioms are by definition unprovable. Godel proved that you cannot prove the consistency of your system of axioms.

>>	Anonymous Sun Oct 13 02:03:55 2013 No.6086407 >>6086325 > Implying nobody studies set theory or foundations of mathematics.

>>	Anonymous Sun Oct 13 02:33:56 2013 No.6086451 File: 3 KB, 400x300, tumblr_mr696b9BGu1szim6vo1_400.jpg [View same] [iqdb] [saucenao] [google] >>6086325 >there are people posting on /sci/ RIGHT NOW that think mathematical logic is just analytic philosophy

>>	marshmellow !Q4j8PCgioI!!o18diCwWpad Sun Oct 13 02:36:25 2013 No.6086452 >Pythagoras gr8 b8 m8

>>	Anonymous Sun Oct 13 02:44:25 2013 No.6086462 > Pythagoras and Euler > be remembered as one of the greatest minds to ever live. Pythagoras and Euler are remembered for having practical applications that every STEM needs. Does Godel have this?

>>	Anonymous Sun Oct 13 02:48:25 2013 No.6086467 File: 717 KB, 1000x793, 1381464686135.png [View same] [iqdb] [saucenao] [google] >>6086325 What about on /g/?

>>	Anonymous Sun Oct 13 02:54:26 2013 No.6086474 >>6086325 lol this its so easy to spot them philosophers are like a more autistic version of mathematicians

Anonymous Sun Oct 13 07:03:07 2013 No.6086635

>>6086243
>I'm only a second year maths student
You hadn't studied this before starting uni?
"Set Theory and the Continuum Problem" will give you a good idea of how Godel's work fits into the maths you are already used to (though you might want to read something like "Numbers, Sets and Axioms" beforehand). That is to say, his (in)completeness theorems will probably not have much effect on you in your mathematical career, but the independence of the continuum hypothesis (which he proved one part of - consistency. Cohen did the other half, showing that the negation of CH is also consistent) will feel "close" to what you have already studied.
For an example of another undecidable proposition even closer to home, try the Whitehead problem in group theory.

>>	Anonymous Sun Oct 13 07:18:06 2013 No.6086649 File: 5 KB, 176x187, 1381000936795.jpg [View same] [iqdb] [saucenao] [google] >hey guys, I used diagonalization to prove some obscure fact that has no practical, and moreover, no serious mathematical implications.

>>	Anonymous Sun Oct 13 07:23:36 2013 No.6086653 >>6086649 how can you say that! don't you know about muh strange loop. haven't you read penrose? godels theory proves that conscious is untangibale and effervescent.

>>	Anonymous Sun Oct 13 07:29:08 2013 No.6086656 Cohen > Godel

Anonymous Sun Oct 13 07:47:37 2013 No.6086677
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>>6086013
You can't prove the consistency of a theory able to talk about natural numbers (e.g. foundational theories like the standard set theories) using weaker systems, like Peano arithmetic.
But I essentially agree with
>>6086649
However, what are "serious mathematical implications" anyway?
One can say that Gödels work had a huge historical impact on the development of computers, so in this sense it has implications.
There are also hands on undecidable problems: Mortal matrix problem: There are certain pairs of integer square matrices, I think of dimension 45, for which you can not decide wether you can multiply them (repetitions allowed) such that they become the 0-matrix.
The post correspondence problem is also a pretty one.

>>6086011
>it said we couldn't prove everything that was true.
Using the word "true" when talking with someone without background on semantics vs. syntax is dangerous.
In any case, you don't need the word "true" to speak of Gödel's incompleteness theorems anyway. They are syntactical statements.

>>6086038
>It basically says that although there are axioms (undoubtedly true statements) that we use to prove things, we cannot necessarily prove all our axioms are true.
What you want to say here is "sound", not "true".

>>6086117
>There exist theorems that cannot be proven formally but are nevertheless true.
This is an oxymoron, a theorem is a proven statement.

>>6086179
Theories are incomplete, not the proofs.

>>I figure that you could create some logic that transcends the logic those theorems uses and then nothing they'd say would apply.
Not if it contains the theorems for peano arithmetic, and hence the truths of the standard interpretation of the natural numbers.

>>6086403
Formally speaking (sorry), axioms are per definition provable from themselves.

>>	Anonymous Sun Oct 13 07:58:06 2013 No.6086689 >>6086677 lol nerd

Anonymous Sun Oct 13 07:59:08 2013 No.6086692

>>6086677
>"serious mathematical implications"
Cauchys work on complex numbers, Galois work on groups, Hilberts work on geometry. These guys are worth remembering in a thousand years. Their work already has had profound implications for both mathematics and sciences. I don't think the problem of multiplying some matrices to zero is a worthy application of Gödels work. Of course, I still feel that his work is just as original, if not more, as the works of the three mentioned above.

>>	Anonymous Sun Oct 13 08:01:07 2013 No.6086695 >>6086692 >Galois work on groups he worked on fields, using groups, he added little to group theory

Anonymous Sun Oct 13 08:07:07 2013 No.6086704
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>>6086692
No, I don't agree. E.g just for proving the first incompleteness theorem he introduced his encoding scheme together with the Gödel beta function to implement primitive recursive functions.
http://en.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function

That is to say, whenever you implement a for-loop in a program, you should thank Gödel.

>>	Anonymous Sun Oct 13 08:13:37 2013 No.6086710 Gurt Knödel

>>	Anonymous Sun Oct 13 08:58:38 2013 No.6086749 Bert Curdles

>>	Anonymous Sun Oct 13 09:04:09 2013 No.6086754 >rivaling Euler Holy fuck no.

>>	Anonymous Sun Oct 13 10:33:29 2013 No.6086913 >>6086179 it's more general than that, it applies to any formal system

>>	Anonymous Sun Oct 13 10:35:28 2013 No.6086918 >>6086462 not as much pythagoras

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