claude_moveml
Literotica Guru
- Joined
- Jan 21, 2002
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A good article about Goedel is http://education.guardian.co.uk/Print/0,3858,4175878,00.html
No formal system general enough to include arithmetic (or other construct that can be used for self-referentiality) can be both consistent and complete.
A formal system is a mathematical construct, with a set of simple axioms (facts like a+b=b+a) which can be used as rules to prove theorems (like a+b+c=c+b+a).
Consistent means that you can't prove a false statement. If you could prove that a=a+1 then you have a problem.
Complete means that you can prove every true statement. This seems more difficult, but you could for example have a formal system where you could prove every statement (true or false), which is complete but not consistent.
The proof is basically the statement "this statement cannot be proved".
Suppose it is false. Then you can prove it. But then you have proved a false statement, so your formal system is not consistent.
Suppose it is true. Then you can't prove it. But then you have a true statement which you cannot prove, so your formal system is not complete.
But a statement can only be true or false, so every formal system is either inconsistent or incomplete. And so no formal system can be both consistent and complete.
And generally you want consistency, so the end result is you can't prove everything.
This seems a mathematical problem, with no real consequences. But all scientific descriptions of the universe are essentially formal systems, and so we can never know everything about it. Moreover, in an incomplete formal system (nearly all useful systems are consistent and therefore ) there are unproveable true statements. But we know they are true, so the human brain must be something more than a machine following (albeit complicated) rules.
Maybe there is a deep connection between the self-referentiality of the universe (quantum mechanic's observer, and on a larger scale conscious self-awareness) and the self-referentiality that enables Goedel's theorem.
No formal system general enough to include arithmetic (or other construct that can be used for self-referentiality) can be both consistent and complete.
A formal system is a mathematical construct, with a set of simple axioms (facts like a+b=b+a) which can be used as rules to prove theorems (like a+b+c=c+b+a).
Consistent means that you can't prove a false statement. If you could prove that a=a+1 then you have a problem.
Complete means that you can prove every true statement. This seems more difficult, but you could for example have a formal system where you could prove every statement (true or false), which is complete but not consistent.
The proof is basically the statement "this statement cannot be proved".
Suppose it is false. Then you can prove it. But then you have proved a false statement, so your formal system is not consistent.
Suppose it is true. Then you can't prove it. But then you have a true statement which you cannot prove, so your formal system is not complete.
But a statement can only be true or false, so every formal system is either inconsistent or incomplete. And so no formal system can be both consistent and complete.
And generally you want consistency, so the end result is you can't prove everything.
This seems a mathematical problem, with no real consequences. But all scientific descriptions of the universe are essentially formal systems, and so we can never know everything about it. Moreover, in an incomplete formal system (nearly all useful systems are consistent and therefore ) there are unproveable true statements. But we know they are true, so the human brain must be something more than a machine following (albeit complicated) rules.
Maybe there is a deep connection between the self-referentiality of the universe (quantum mechanic's observer, and on a larger scale conscious self-awareness) and the self-referentiality that enables Goedel's theorem.
