Godel's incompleteness theorems


by Shyan
Tags: godel, incompleteness, theorems
Shyan
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#1
Jan21-13, 01:39 PM
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I'm trying to understand Godel's incompleteness theorems.
But I have a difficulty.
Are they about any set of related prepositions whether mathematical or philosophical or anything?
I mean,imagine someone wants to have a theory describing e.g. a literature concept(Which apparently involves no math). Can we apply Godel's theorems to it?
Another question maybe not so related to the previous ones.Can we say that everywhere in mathematics,there is a set of axioms which serve as the beginning of the reasoning?Is there something in mathematics which people say it is obviously true,we neither prove it,nor assume it?

Thanks
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Sayajin
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#2
Jan23-13, 06:36 PM
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You can't apply Godel's theorem to non mathematical (non scientific) concepts like literature. Literature is not something consistent. The structures they use in literature are not deterministic and well defined. You can basicaly have metaphors and symbols which they think they have another meaning except the dictionary one but there is no transofmation that tells you how to convert metaphor to plain text and the oposite. Most of the time people interpet the meanings of the things accourding to their own believes.

And yes most of the time there are axioms in maths we assume but we can't prove. We use them as a fundamental knowlege to logicaly prove other things.
Charles Wilson
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#3
Jan23-13, 07:26 PM
P: 56
From _Godel's Proof_, Nagel and Newman, ISBN 0-7100-7078-0, p.98:

"[Godel's conclusions]...show that the prospect of finding for every deductive system (and, in particular, for a system in which the whole of arithmetic can be expressed) an absolute proof of consistency that satisfies the finitistic requirements of Hilbert's proposal, though not logically impossible, is most unlikely. They also show that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference."

Consider the following statements:

"Somewhere, over the rainbow, bluebirds sing..."
"The photon simply had to go through one slit or the other..."
"There is a greatest number that can stated in the English language in 19 syllables..."
"There are propositions within arithmetic which are true but cannot be proven true within the rules of arithmetic..."
"There exists a successor to zero..."

Go ahead, spend the day thinking about these statements. There is at least one problem here. "Which of these can be represented by logical symbols within an axiomatic system?" The questions of "Truth" come long after the answer to the axiomatization question and may not "legally" come up at all.

"Godel showed (i) how to construct an arithmetical formula G that represents the meta-mathematical statement: "The formula G is not demonstrable". This formula G says of itself that it is not demonstrable...But (ii) Godel showed that G is demonstrable if, and only if, its formal negation ~ G is demonstrable..."

G'sP, p. 85.

This is all heavy slogging through a blizzard but it "demonstrates" that the proof is of Axiomatic Systems. Whatever else might apply to poetry is problematic but not excluded from thought. Focus first, however, on Godel's Domain in terms of axiomatic systems.

To your last question: When I taught HS Geometry, our school used the UCSMP "system". This text opened up with a coupla' sections on "What is a point?". The kids would go nuts. "This is STOOPIT!!" No. It's not. For some, though, everything had to be argumentative - without the "Logic" and "Rhetoric".

It may take time but allow yourself some freedom to accept that, "There might be more than one line through a given point parallel to a given line". But maybe not, as well. "...And what is a point, anyway?"

You've asked the right questions.

Look at Feynman graphs. "Two sideways "Vs" connected by a squiggly line! Simple!" Yes, and very deep.

Hope this helps.

Charles

Shyan
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#4
Jan24-13, 01:08 AM
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Godel's incompleteness theorems


Well,looks like I didn't give a good example.
I meant can we apply godel's theorems to systems which don't use mathematics,like a philosophical system?
And about my other question,I didn't mean axioms.I said sth we neither prove,nor assume.
Thanks
pwsnafu
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#5
Jan24-13, 01:51 AM
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Quote Quote by Shyan View Post
Well,looks like I didn't give a good example.
I meant can we apply godel's theorems to systems which don't use mathematics,like a philosophical system?
No, Godel's theorem requires a very specific axiomatic system: basic arithmetic. If your system doesn't contain that, Godel does not apply.
Charles Wilson
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#6
Jan24-13, 10:28 AM
P: 56
Well, I thought I was being helpful.
I'm sorry I didn't understand, it appeared to be "right up my alley".
I'll try to do better next time.

CW
Shyan
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#7
Jan24-13, 12:47 PM
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No charles,I just hadn't seen your explanation when I was writing that post.
But I should confess that I didn't understand much and decided to delay such discussions until I study more about the subject.
Thanks all guys


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