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Comment by Grue3 | Hacker News Reader

It's just plain wrong to say that there are more facts in a theory than proofs of facts. Both of these sets are countably infinite. Godel's theorem proves that provable facts are a strict subset of all facts.

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While I agree with you, in the context of a "hand wavey" explanation, "more" is not rigorously defined and so saying "plain wrong" is silly. A strict subset relationship is a reasonable interpretation of "more" in the context of an informal discussion.

Leaving aside the fact that talking "informally" about infinities easily leads to wrong conclusions, the OP wasn't even talking about a strict subset relationship. The comparison was between a set of proofs and a set of facts, which are sets of different mathematical objects. So at best you can say that each proof proves a fact (a fact can have several proofs) and the set of facts that have a proof is a strict subset of all facts. Or, informally, there are more facts than facts which can be proved. Oh, wait, this just restates the theorem we're trying to prove in the first place and doesn't give any insight into why it's correct.

Yes, but the use of "which in turn" suggests that was not meant; it suggests that what follows is a real deduction and not a mere rephrasing.

>It's just plain wrong to say that there are more facts in a theory than proofs of facts. Both of these sets are countably infinite.

Some infinities are larger than others. This was proved by Georg Cantor, who created set theory.

And if you read about Cantor you will find he defined a term called 'countably infinite'. No 'countably infinite' set has a higher cardinality than any other.

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