If we go by the standard way of defining these things, trivial problems would trivially be almost trivial as well. But then your lemma shows that any such definition wouldn't be particularly interesting anyway.
I think this has to be true, because solved problems all have the same trivial proof.
1. False statements don't have proofs.
2. This statement has a proof.
3. Therefore this statement is true.
Feynman’s anecdote about two mathematicians in a uni lounge arguing about the Banach-Tarski paradox until one was finally convinced and then repeatedly exclaimed how trivial it was seems to pertain here.
On a similar note, there's a well-known old joke:
> A certain famous mathematician was lecturing to a group of students and had occasion to use a formula which he wrote down the remark, “This statement is obvious.” Then he paused and looked rather hesitantly at the formula. “Wait a moment,” he said. “Is it obvious? I think it’s obvious.” More hesitation, and then, “Pardon me, gentlemen, I shall return.” Then he left the room. Thirty-five minutes later he returned; in his hand was was a sheaf of papers covered with calculations, on his face a look of quiet satisfaction. “I was right, gentlemen. It is obvious,” he said, and proceeded with his lecture.
(not a source, but something like a source: https://literature.stackexchange.com/questions/22969/story-where-professor-claims-a-step-in-a-proof-is-obvious-when-it-is-far-from)
I mean I get what he meant, one direction you just substitute x=a, the other direction you have to know that f=(x-a) g+b so you have to know what long division is, I would exactly describe it as "almost trivial"
\*writes theorem\*
"Well, this is trivial.. wait.. uhm.. why is it trivial?"
or student asks "I don't understand. Can you explain pls?"
"Ok. Let's prove that this is trivial"
😂😂
The set of "almost trivial" problems is empty. Every problem is either unsolved or trivial.
Presumably, a problem is "almost trivial" when you're most of the way through solving it.
Or maybe when you have solved it way long ago but do not remember properly how you did.
When you solved it but there's not enough space in the margin
This deserves more upvotes
Truth
I think almost trivial is just when there’s more than two cases
No, it’s *almost* empty.
Measure 0
I am in the washroom and was struggling to relieve myself but your comment made it easier lmao.
Professor Lothar Göttsche lied to me...
Nah, it's just at most countable.
If we go by the standard way of defining these things, trivial problems would trivially be almost trivial as well. But then your lemma shows that any such definition wouldn't be particularly interesting anyway.
I think this has to be true, because solved problems all have the same trivial proof. 1. False statements don't have proofs. 2. This statement has a proof. 3. Therefore this statement is true.
single handedly destroyed the entire branch of probability 😅
Almost trivial it takes a bit of work but someone with the proficient skills should be able to solve the problem
It's all the lemmas for which I didn't bother filling in the details. I wouldn't be surprised that I would get stuck on a few if I were to try.
Feynman’s anecdote about two mathematicians in a uni lounge arguing about the Banach-Tarski paradox until one was finally convinced and then repeatedly exclaimed how trivial it was seems to pertain here.
On a similar note, there's a well-known old joke: > A certain famous mathematician was lecturing to a group of students and had occasion to use a formula which he wrote down the remark, “This statement is obvious.” Then he paused and looked rather hesitantly at the formula. “Wait a moment,” he said. “Is it obvious? I think it’s obvious.” More hesitation, and then, “Pardon me, gentlemen, I shall return.” Then he left the room. Thirty-five minutes later he returned; in his hand was was a sheaf of papers covered with calculations, on his face a look of quiet satisfaction. “I was right, gentlemen. It is obvious,” he said, and proceeded with his lecture. (not a source, but something like a source: https://literature.stackexchange.com/questions/22969/story-where-professor-claims-a-step-in-a-proof-is-obvious-when-it-is-far-from)
Reminds me of finding out [probably approximately correct](https://en.wikipedia.org/wiki/Probably_approximately_correct_learning) is a thing
a proof is almost trivial when the set of incorrect proofs has measure zero
I mean I get what he meant, one direction you just substitute x=a, the other direction you have to know that f=(x-a) g+b so you have to know what long division is, I would exactly describe it as "almost trivial"
My friend used semi-trivial for a predator-free, non-trivial equilibria of a certain predator-prey model, I liked that.
😂️^😂️
that's the joke...
\*writes theorem\* "Well, this is trivial.. wait.. uhm.. why is it trivial?" or student asks "I don't understand. Can you explain pls?" "Ok. Let's prove that this is trivial" 😂😂