# Is there a precise definition for “follows directly that”?

I've taken some thought before asking this question as I thought it might be opinion-based, but I arrived at a formulation which I think is adequate for MSE. Therefore, here it is:

I've always had a problem with the statement "it is easy to see that...". This is something that not only is personal, but also time-dependent.

But I think that "it follows directly that..." is a statement that is more sound. Intuitively, I would say that "$X$ follows directly from $Y$ if we can go from $Y$ to $X$ by means of definitions only, i.e., with no theorems". Of course this is far from good, but encapsules the following idea:

One would like to say that, having defined that in a metric space:

$\lim x_n =x :\iff \forall \epsilon >0 ~\exists N :n >N \implies d(x_n,x)<\epsilon$

it follows directly that:

$\lim x_n=x \iff d(x_n,x) \rightarrow 0$

Of course, if we allow my naive definition to hold, anything that can be demonstrated would follow directly from the axioms (by allowing theorems to enter the proof sneakly in the form of conclusions). Therefore, I think it would be a good idea to refine the naive definition in the following way:

"$X$ follows directly from $Y$ if we can go from $Y$ to $X$ by means of definitions only, i.e., with no theorems, in less than an specified number $N$ of implications (say $N=5$, for example)".

I have no substantial knowledge of logic, so I would like to ask for an answer that doesn't assume much knowledge of the subject. But it is okay if that's not possible. The question is:

Is there a way to formalize the naive concept of "follows directly"?

PS: What I mean by a "theorem" is simply a statement labeled as "theorem" in a given text, for example.

• LOL, I'm feeling like my comment to you yesterday occasioned this post. I had it in mind that it should follow in a line or two from definitions, but I should probably go back and address your concerns ... – Ted Shifrin Sep 18 '15 at 0:25
• @TedShifrin Haha! Yes, it did! But not in a bad way! I managed to arrive at a conclusion about that, but I must admit that it was not immediate for me (I danced with the fact that geodesics minimize length locally etc). In fact, if you have a more direct proof, I would gladly accept it. : ) – Aloizio Macedo Sep 18 '15 at 0:32
• Maybe the terms 'obvious', 'easy to see', 'follows directly,' etc. all mean 'it's not really that hard and I want you to figure it out for yourself'. – BruceET Sep 19 '15 at 21:39