Limitations of Proving Theorems: Is Infinity a Barrier?

[cragar]

If a theorem required an uncountable amount of information or symbols to prove it, would this mean it could be true but unprovable. Are we just limited because we can only write a countable number of symbols? Could the theorem be proved in some other sense?

 

[haruspex]

Necessarily, a proof is of finite length. Of course, it may be unprovable in one formal system for this reason, but provable in a more powerful one.

 

[cragar]

so in a certain model that theorem might require an infinite amount of information to prove it, but in another more powerful model it might only need a finite amount of information to prove it. When you say more powerful, do you mean different axioms of more axioms.

 

[haruspex]

so in a certain model that theorem might require an infinite amount of information to prove it, but in another more powerful model it might only need a finite amount of information to prove it. When you say more powerful, do you mean different axioms of more axioms.

If different axioms it would be effectively a different theorem. It would have to be added axioms.

 

[cragar]

ok, so If i had a theorem that required an infinite amount of information to prove it, but then if I just took that as an axiom I could could compress an infinite amount of information into a single finite statement.

 

[haruspex]

Maybe, but I'm doubtful about the very concept of a theorem requiring an infinite proof. Proofs are not infinite by definition. All you can say is that the proposition is neither provable nor disprovable within the system, in which case, yes, you can add it (or its negation) as an axiom.

 

[lugita15]

Maybe, but I'm doubtful about the very concept of a theorem requiring an infinite proof. Proofs are not infinite by definition.

Well, at least normally. But look at this.