Proof the Godel's Incompleteness Theorem

In summary, the Goedel's Incompleteness Theorem states that any system of axioms is either inconsistent or incomplete. This means that there will always be some statements within the system that cannot be proven or disproven. This applies to all logical systems based on axioms, including mathematics, but not to other sciences that rely on experimental evidence. It is possible that an inconsistent axiomatic system used in physics could affect the mathematical foundation, but there is no evidence that any important mathematical system is inconsistent. Theoretical physics may have theories created before experimental evidence, but the experimental evidence is still valid.
  • #1
newton1
152
0
who can show me how to proof the Godel's Incompleteness Theorem ?
it that mean the system of mathematics will never be complate??
 
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  • #2
I doubt that anyone can show you how to prove Goedel's incompleteness theorem- its very long and very deep. There are a number of books written on it alone.

Yes, Goedel's theorem says that any system of axioms (large enough to encompass the natural numbers) is either inconsistent (in which case it's useless) or incomplete (in which case it's not perfect!).

Saying that a system is incomplete means there exist some theorem, statable in terms of the system which can be neither proven nor disproven. Of course, you could always add that theorem itself as an axiom but then you would have some other theorem that can be neither proven nor disproven.

One cannot prove, in absolute terms, that most of the systems used in mathematics are consistent- it is possible to prove, for example, that Euclidean geometry is consistent if and only if hyperbolic geometry is, or that Euclidean geometry is consistent if and only if algebra is, which is consistent if and only if the natural numbers are, which is true if and only if set theory is consistent...

Since there are one heckuva lot of things we don't know HOW to prove, most mathematicians are willing to live with the knowledge that they can't prove EVERYTHING!
 
  • #3
i see...
are this theorem include the other science system like physics system??
 
  • #4
It applies only to logical systems based on axioms. That includes all of mathematics. It does not apply to physics or other sciences that are based on experimental evidencel.

It is conceivable that, if a particular axiomatic system used to model physics were inconsistent, the mathematical foundation would be in trouble. The experimental evidence, of course, would still be valid. I will point out that there is no evidence that any important mathematical system is inconsistent.
 
  • #5
what i mean is theoretical physics...
some theory was create by scientist before they have the experimental evidence
 
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What is Godel's Incompleteness Theorem?

Godel's Incompleteness Theorem is a mathematical theorem that was proven by Austrian mathematician Kurt Godel in 1931. It states that in any formal axiomatic system that is sufficiently complex, there will always be true statements that cannot be proven or disproven within that system.

Why is Godel's Incompleteness Theorem important?

Godel's Incompleteness Theorem has had a significant impact on the fields of mathematics, logic, and computer science. It has shown that there are limits to what can be proven within a formal system and has raised questions about the foundations of mathematics and the nature of truth.

What is an example of a true statement that cannot be proven within a formal system?

One example is the statement "This statement is unprovable." If this statement were provable, it would be false. But if it is false, then it must be provable. This creates a paradox, demonstrating that the statement cannot be proven within the system.

Can Godel's Incompleteness Theorem be applied to other areas besides mathematics?

Yes, Godel's Incompleteness Theorem has been extended to other areas such as logic, computer science, and even philosophy. It has implications for any system that relies on axioms and rules for determining truth.

Does Godel's Incompleteness Theorem disprove the idea of a complete and consistent theory of everything?

Godel's Incompleteness Theorem does not necessarily disprove the idea of a complete and consistent theory of everything. It simply states that such a theory cannot be proven within a formal system. It is possible that there could be a larger system in which a theory of everything could be proven, but this is currently unknown.

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