? about Godel's incomplete theorm

  • Thread starter raven1
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In summary, the theory of real fields is incomplete and does not apply to Euclidean geometry. It is complete for fields that are extensions of the real numbers. Math itself is flawed because there are statements that can neither be proven nor disproven.
  • #1
raven1
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I read the theorm and the proofs, but what i am unclear about is when is it used or what statements are unprovable. as far as i know it doesn't apply to Euclidean geometry so what branchs of math does it apply. part of the reason i am asking is on another website i seen someone try to use this theorm to show that .99... does not equal 1 and math itself is very flawed , i felt that the theorm is being misused but i probably shouldf learn more before i say anything
 
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  • #3
First, it is "incompleteness" theorem, not "incomplete" theorem. The theorem itself is complete. Essentially, it says that in any axiom system large enough to include the natural numbers is "incomplete"- that is that there exist statement which can neither be proven nor disproven. The proof of that theorem itself "constructs" a statement that can be proven nor disproven but in a very abstract way- it uses a way of assigning numbers to statements (the Godel numbering) such that the statement assigned a certain number be shown to neither provable nor disprovable- but doesn't tell us exactly what that number is and certainly not what the theorem is. Indeed, given a statement that cannot be proven nor disproven, we could take that statement (or its negation) as an axiom and have a new system- which would then have a new statement that could not be proved nor disproved.

I can't imagine that having anything to do with 1= 0.999... That follows immediately from the definition of the base 10 numeration system.
 
  • #4
As either Halls or Hurkyl is very keen to point out, the theory of Real Fields is complete and not remotely susceptible to this line of attack. Please post a link to this person's attempted disproof; I'm sure many people here would like to point out their egregious error.
 
  • #5
raven1 said:
I read the theorm and the proofs, but what i am unclear about is when is it used or what statements are unprovable. as far as i know it doesn't apply to Euclidean geometry so what branchs of math does it apply. part of the reason i am asking is on another website i seen someone try to use this theorm to show that .99... does not equal 1 and math itself is very flawed , i felt that the theorm is being misused but i probably shouldf learn more before i say anything
First off, godel's theorem talks about the existence of such a statement. It doesn't tell us what they are. Finding one/showing that a statement is undecidable is a lot harder. Examples: The Continuum hypothesis was shown to be undecidable within ZFC. I also know that the Goodstein's thoerem is an undecidable statement within peano arithmetic, however it can be proven with a stronger system. Whoever used godel's thm. to show that .999... does not equal 1 is 99.999999% probably a crackpot that has no idea what they are talking about.
 
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  • #6
matt grime said:
As either Halls or Hurkyl is very keen to point out, the theory of Real Fields is complete and not remotely susceptible to this line of attack. Please post a link to this person's attempted disproof; I'm sure many people here would like to point out their egregious error.

I started to ask for a link but then it occurred to me that I waste entirely too much time on that sort of thing already!
 

What is Godel's Incompleteness Theorem?

Godel's Incompleteness Theorem is a mathematical proof, developed by Kurt Godel in 1931, that states that in any formal mathematical system, there will always be statements that are true but cannot be proven within the system itself. In other words, there will always be limitations to what a mathematical system can prove.

How does Godel's Incompleteness Theorem relate to logic and mathematics?

Godel's Incompleteness Theorem is relevant to logic and mathematics because it shows that there are inherent limitations to these fields. It challenges the idea that all mathematical truths can be proven through a set of logical rules, and instead highlights the importance of intuition and creativity in mathematical thinking.

What are some real-world implications of Godel's Incompleteness Theorem?

Godel's Incompleteness Theorem has had a significant impact on the field of mathematics, as it has forced mathematicians to rethink their approach to proving mathematical truths. It has also influenced other fields, such as computer science and artificial intelligence, as it demonstrates the limitations of formal systems and the importance of human intuition in problem-solving.

How was Godel's Incompleteness Theorem received by the mathematical community?

Initially, Godel's Incompleteness Theorem was met with skepticism and resistance from some members of the mathematical community. However, as more mathematicians studied and verified the proof, it became widely accepted and is now considered one of the most significant results in the history of mathematics.

Are there any criticisms or counterarguments to Godel's Incompleteness Theorem?

While Godel's Incompleteness Theorem is widely accepted, some have argued that it may not apply to all formal systems or that it may have limitations in its own proof. Others have also questioned the implications of the theorem for the concepts of truth and knowledge. However, these criticisms have not been able to disprove the theorem itself.

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