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January 22nd, 2019, 10:57 PM   #1
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Gödel's 2nd theorem ends in paradox

Godel's 2nd theorem ends in paradox

Godel's 2nd theorem is about

"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”

But we have a paradox

Gödel is using a mathematical system
his theorem says a system cant be proven consistent


Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
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January 23rd, 2019, 10:35 AM   #2
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Yes. So assume you can prove within the system that the system is consistent and complete.

Either two possibilities:
- Either the system is inconsistent. Done
- The system is consistent. Then Godel applies. Contradiction.

So whatever the system, if you can write a proof of consistence and completeness, the system must be inconsistent. Not a paradox, just very counterintuitive.
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