Limitations of Proving Theorems: Is Infinity a Barrier?

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Discussion Overview

The discussion centers around the limitations of proving theorems, particularly in relation to the concept of infinity. Participants explore whether a theorem could be true but unprovable due to the requirement of an uncountable amount of information or symbols, and how different formal systems might affect provability.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that a theorem requiring an uncountable amount of information could be true but unprovable due to limitations in symbol representation.
  • Others argue that while a proof is necessarily of finite length, it may be unprovable in one formal system but provable in a more powerful one.
  • A participant questions whether "more powerful" refers to different axioms or additional axioms, suggesting that different axioms could lead to effectively different theorems.
  • Another participant suggests that if a theorem requires infinite information to prove, it could be treated as an axiom, thus compressing that infinite information into a finite statement.
  • Some express doubt about the concept of a theorem requiring an infinite proof, asserting that proofs are finite by definition and that a proposition could be neither provable nor disprovable within a system.
  • One participant reiterates skepticism about the idea of infinite proofs, emphasizing that proofs are not infinite by definition, though they acknowledge exceptions in certain contexts.

Areas of Agreement / Disagreement

Participants express a range of views, with no consensus reached on the implications of infinity in theorem proving. Some agree on the limitations of formal systems, while others challenge the notion of infinite proofs.

Contextual Notes

The discussion highlights the dependence on definitions of proofs and the implications of different axiomatic systems, with unresolved questions about the nature of infinity in mathematical proofs.

cragar
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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?
 
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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.
 
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.
 
cragar said:
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.
 
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.
 
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.
 
haruspex said:
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.
 

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