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> ... and could have zillions of equally uninteresting variations. Which themselves could generate uninteresting theorems: Theorem: The set of such uninterest
by cgriswald 5y ago
> ... and could have zillions of equally uninteresting variations.
Which themselves could generate uninteresting theorems:
Theorem: The set of such uninteresting variations of "All natural numbers are larger than minus ten" is countably infinite.
Proof: Let the uninteresting variations of "All natural numbers are larger than minus ten" be represented as "All natural numbers are larger than minus n" where n is a natural number. Since there exists a one-to-one correspondence between the natural numbers and the natural numbers, the set of uninteresting variations of "All natural numbers are larger than minus ten" is countably infinite.∎
- DenisM 5y agoThe set of all theorems, or even all statements, is countably infinite. The symbolic representation of a statement is only a string of characters and can be converted to a very large integer. Even our notion of uncountable sets is itself countable, in that we use language to reason about them. It actually pretty sad - we’re are two-dimensional creatures in the three-dimensional world.