Let's say you just got a job as Enumerator at a company that sells integers, and not more than an hour into your first day the vice president of Parity (E. Stevens) tells you "We need something completely new. People like the product, they're comfortable with it, but you sit on your ass in this business and you end up like the roman numeral guys. I want an integer that no one's even thought of before, and I want it yesterday.''
For a moment you might panic. Now that you think about it, there really are a lot of integers "in circulation,'' especially with all those computers out there. Every transaction, web search, and algorithm is full of them! You fume: "if only someone kept track of these things I could use an Inducter..."
Soon enough you'd settle down and realize that with a little ingenuity a solution is not so hard to come by. One just needs a suitably large number (say, larger than ((100!)!)!) constructed somewhat haphazardly. In no time you'd have a candidate impressive enough for Mr. Stevens. (In fact, why not try it before reading on?)
I hope you agree that writing out an integer no one's ever seen is at first a bit exhilarating, like skiing through fresh snow. But it gets old fast: Z is inexhaustible, and you could conceivably churn out newfangled numbers forever.
Perhaps a more interesting exercise is to think of numbers that have not only never been seen but never could be: the square root of 2, for example. In fact, any irrational number will do since by definition they all have infinite decimal expansions (so you'll never catch a glimpse of their last digit). We can take it one step further with the transcendental numbers first defined by Euler. They're irrational, too, which makes them "unseeable" in that way, but we also know that despite there being infinitely many, mathematicians have only ever found a handful (e, π, and a few others) "in the wild.'' Spotting a transcendental—and then proving your find—is a lot like finding a needle in a haystack filled with spiders: it's very difficult.
So now we have two classes: the set of "unseen" numbers and the "transcendental" ones. Both are slippery—the former because anytime we put our finger on one it no longer qualifies, and the latter because we can't seem to think of any.
What if we took the worst of both worlds: a set of things such that (a) even finding one is a real achievement and (b) once you find one, you have to throw it out. Can you even conceive of such a set?
Sure: let's call it "the set of inconceivable things." In it is all the stuff that can never be observed, deduced, or imagined. Even things we cannot know, like the last digit of π, or whether porcupines would work better in C++ or Python, or which eigenvalue will pop out when a wave function collapses, are at least thinkable.
Yet I have a very strong hunch that there are plenty of things that aren't, things we cannot even think of thinking about. What's more, there could be far more of them than their conceivable counterparts! The question is, how important are they?