User blog comment:Kyodaisuu/Pseudo Rayo's number/@comment-119611-20150102151948/@comment-1605058-20150102154105

Very well said, I agree with this completely. I was myself considering if we shouldn't constrain ourselves to definitions which provably (in some agreed upon system like ZFC) specify exactly one number (or another object), i.e. we can prove that there is only one number N satisfying the definition. To not get into details, the reason for this is that if we cannot prove something, then there are models in which it's true or false, of our choice.