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If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will n

by wbhart 10y ago

If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. So I can assume them to be true or false, and develop perfectly consistent mathematics. In this way, I can always add more rules to the system (albeit a nonstandard one).

lurker83256 10y ago
This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).
- baddox 10y ago
  Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
  - drdeca 10y ago
    I think by adding the axiom, they mean considering the axiom system with the old axioms, and also the one being added? So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?
baddox 10y ago
A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.
lmm 10y ago
> If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. Assuming the original system was consistent (which you can never prove), yes. But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).

by wbhart 10y ago

lurker83256 10y ago

This is a misunderstanding. Statement A is actually true in the system, you just cannot prove that it is true. Adding an axiom specifying its falsity would be a contradiction (although you could not prove this).

baddox 10y ago
Adding either the Godel sentence or its complement would ruin the consistency of the axiomatic system, because the whole point of the Godel sentence is that it claims that itself cannot be proven to be true in its axiomatic system. But you don't get to add axioms to an axiomatic system anyway, because doing so yields a new axiomatic system to which the original Godel sentence does not refer.
- drdeca 10y ago
  I think by adding the axiom, they mean considering the axiom system with the old axioms, and also the one being added? So, assuming the initial system is consistent, adding the godel sentence, or its negation, produces a new consistent system, iirc. But in the second case, it would be omega-inconsistent ?

baddox 10y ago

A Godel sentence claims that itself cannot be proven in its axiomatic system. If you add another axiom, you have a new axiomatic system that can prove the Godel sentence in the original axiomatic system, but the new axiomatic system will have its own new Godel sentence.

lmm 10y ago

> If they can neither be proved true or false within the system, that means their truth or falsity can be added as axioms to the system and a contradiction will never be reached. Assuming the original system was consistent (which you can never prove), yes. But that doesn't help you, because the new system with these additional axioms will still, provably, contain statements that can't be proven true or false within that system (or else be inconsistent). (Turing did try some programme where you keep adding new axioms and take the transitive closure over all the axioms you'd add, but AIUI that doesn't work out either).