1 Answer
A theorem, by definition, is a statement that has been proven to be true based on a set of axioms or previously established theorems. So, once a theorem has been proven, it cannot be false within the system or framework in which it was proven.
However, a theorem might be considered false if:
In other words, the theorem is only true as long as the logic and assumptions used to prove it are correct. If either of those are wrong, the theorem can’t be considered true anymore. But a proven theorem itself, when it's proven correctly, can't suddenly be false—it’s the assumptions or proof that might be the issue.
However, a theorem might be considered false if:
- The axioms or rules of the system it's based on are later found to be incorrect or incomplete.
- The proof of the theorem is flawed or invalid.
- It turns out the theorem only holds in specific cases, and not in general.
In other words, the theorem is only true as long as the logic and assumptions used to prove it are correct. If either of those are wrong, the theorem can’t be considered true anymore. But a proven theorem itself, when it's proven correctly, can't suddenly be false—it’s the assumptions or proof that might be the issue.