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Trivial watch

1. Since starting my actuarial exams, and when doing some of the later modules at A level, I found the words "obvious", "trivial", "clear" etc. appear in more and more of my maths. While it seems obvious (there it is again) when to use these words, I've never really thought about what these words actually mean, or whether they mean different things.

I'm sure others on this forum use these words a lot; what is your interpretation of them.
2. Not that anyone uses these words terribly carefully, and they are often used interchangeably, but here are my thoughts on when they mean different things:

Obvious: Intuitively true, but may not be easy to prove, E.g. "2+2 = 4", or "every closed curve in R^2 that has no self intersections divides the plane into two regions". In extremis, might not even be true: It's "obvious" that you can't decompose a sphere into two spheres of the same size, but it isn't true (if you assume the axiom of choice).

Clear: Something that a mathematician (who's sufficiently experienced in the area) can tell is going to be true and will know how to prove. The proof might be simple, or it might be lengthy, but it shouldn't involve any large intuitive leaps.

Trivial: Something a mathematician can see is true and can prove in a straightforward fashion without much thought.

The distinction between 'Clear' and 'Trivial' is particularly subjective: what one mathematician thinks is trivial might not even be clearly true to someone not as good in the subject.
3. (Original post by Drederick Tatum)
Since starting my actuarial exams, and when doing some of the later modules at A level, I found the words "obvious", "trivial", "clear" etc. appear in more and more of my maths. While it seems obvious (there it is again) when to use these words, I've never really thought about what these words actually mean, or whether they mean different things.

I'm sure others on this forum use these words a lot; what is your interpretation of them.
"Obvious": 'immediate consequence of the definitions', or 'something that seems like it should be intuitively true'.
"Trivial": when used properly, this should mean something like 'the simplest case' or 'the emptiest case' (e.g. the trivial product, the trivial group). It can also mean 'a simple application of an earlier theorem'. Often it's just used to mean 'I find this quite easy', though.
"Clear": 'the proof is straightforward and shouldn't involve any big leaps'.
4. I guess this is tangentially-related. While I have nothing to say on this, this reminded me of something I recently read in Hardy's A Course of Pure Mathematics (10th ed) that I thought might be interesting: they're remarks by Littlewood on the word "almost obvious".

Instead of paraphrasing what he said here, I'd thought I'd get a Google link to share, but then I thought an archive.org link would be better since Google links have a habit of expiring. However, archive.org only has earlier editions and it's curious to see how they differ:

Earlier edition

Later edition
5. (Original post by DFranklin)
In extremis, might not even be true: It's "obvious" that you can't decompose a sphere into two spheres of the same size, but it isn't true (if you assume the axiom of choice).
Sorry, but I don't understand what your saying. I know what the Banach-Tarski paradox is but what you said seems wrong.

Firstly, are you trying to say that its not true when you don't assume AC? Because, hasen't it been proven the ZF+AC is consistent if ZF is consistent? So I don't see how you can prove something true using AC and then that not being true in ZF. Has Banach-Tarski paradox actually been proven to be independent of ZF?
6. The existence of non-measurable sets is closely linked to the axiom of choice. Just because something is true in one set of axioms doesn't mean it is true in all.

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