Many times I hear physicists say "$x$ is finite" when what they really mean is $x>0$ or $x\neq0$.

Why is that?

Zero is a valid finite number. What's wrong with simply saying $x$ is strictly positive?

Since $\infty$ is a mathematical construct which has nothing to do with our physical measurements (it can be used as an abbreviation sometimes in order to say something more subtle, but no measurement will ever yield $\infty$), what sense is there in the distinction between finite and infinite quantities in physics anyway?

• This is not just a physicists' convention. From Wikipedia on the definition of a finite number: In mathematical parlance, a value other than infinite or infinitesimal values and distinct from the value 0. – Qmechanic Oct 18 '16 at 12:29
• 1) I agree that is bad practice to adopt the terminology that you refer. 2) No measurement will ever yield an infinite outcome, but there are measurements that do not yield an upper bound as well (e.g. the total energy of a some systems can be arbitrarily increased). It is not wrong in my opinion to consider the spectrum of the associated observables unbounded, i.e. "going up to infinity". – yuggib Oct 18 '16 at 12:43

Many interesting quantities in physics are actually ratios, proportionality factors, slopes ... and it is often more or less arbitrary whether the quantity you find in textbooks is one number or its reciprocal. So ratios of $a:0$ or $0:a$ will both be special (and will often require equally special treatment) and you often need a word to denote the usual case that is neither of these.