Godel's incompleteness theorems

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In summary, Godel's incompleteness theorems show that there are statements in mathematics which are true but cannot be formally proven.
  • #1
ShayanJ
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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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  • #2
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 knowledge to logicaly prove other things.
 
  • #3
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
 
  • #4
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
 
  • #5
Shyan said:
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.
 
  • #6
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
 
  • #7
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
 

Related to Godel's incompleteness theorems

1. What are Godel's incompleteness theorems?

Godel's incompleteness theorems are two famous mathematical theorems that were proven by mathematician Kurt Godel in the 1930s. They state that in any formal axiomatic system that is powerful enough to represent basic arithmetic, there will always be statements that are true but cannot be proven within the system. This means that there will always be gaps or limitations in our understanding of mathematics and logic.

2. How do Godel's incompleteness theorems relate to the foundations of mathematics?

Godel's incompleteness theorems shook the foundations of mathematics by showing that no matter how well-constructed a system of axioms and rules may seem, it will always have inherent limitations. This means that there will always be statements that are true but cannot be proven within the system, challenging the idea of a complete and consistent set of axioms as the basis of mathematics.

3. What is the significance of Godel's incompleteness theorems?

Godel's incompleteness theorems revolutionized the field of mathematics and had a profound impact on logic, philosophy, and computer science. They showed that there are inherent limitations to what can be proven and known within any formal system, and that there will always be gaps in our understanding. This has led to new developments and approaches in these fields, such as the study of unprovable statements and the use of alternative logical systems.

4. How did Godel prove his incompleteness theorems?

Godel's proof of his incompleteness theorems was a complex and groundbreaking mathematical feat. He used a technique called "Godel numbering" to assign numbers to symbols and statements within a formal system. He then constructed a statement that essentially says "this statement cannot be proven within the system." If the statement is true, then it cannot be proven within the system, and if it is false, then it can be proven. This creates a paradox, proving the limitations of the system.

5. Are there any potential criticisms or challenges to Godel's incompleteness theorems?

While Godel's incompleteness theorems have been widely accepted and have had a profound impact on mathematics and other fields, there have been some criticisms and challenges. Some argue that the theorems only apply to formal systems and do not necessarily reflect the limitations of human reasoning. There have also been attempts to refute the theorems or find loopholes in the proofs, but so far, no definitive counterarguments have been accepted by the mathematical community.

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