>Are there things that we can't test that do exist?
There's many reasons to believe that objects that exit our light cone continue to exist after they do, even though they could never have any future interaction with us to confirm that. (Say a spaceship leaves Earth at near the speed of light in a straight line, and then enough time passes that the space between the ship and Earth is expanding so fast that the spaceship or any kind of signal from the spaceship would have to travel faster than light to return to Earth, which is impossible. Believing that the spaceship disappears when it exits our light cone requires believing in unnecessarily more complicated physics.)
Scientifically speaking, no. A scientific hypothesis must be falsifiable, and to be falsifiable it must be testable. I guess in some sense you could claim that there are hypothesis that are testable, but which we do not have the capacity to test. But then, is the claim that "one day in the future, we will be able to test this other claim" itself falsifiable? I'd argue not (it's a recognizable, not decidable claim, in the computational sense, and I think that for a claim to be falsifiable, it must be decidable).
This is certainly one understanding about what science should be (although not a scientific one interestingly enough). Personally I prefer Thomas Kuhn's demarcation, which by my understanding concentrates more on whether a scientific program is producing interesting predictions which turn out to be true.
I'm not speaking about science as a whole or a scientific program, but a scientific claim. CERN is certainly not falsifiable, but it produces predictions which (often) turn out to be true. It does so by devising falsifiable claims and then testing those claims.
In other words, the method to create interesting predictions which turn out to be true is to create interesting predictions, then test those predictions, and update your understanding of the world based on them. Once your world-model is good enough, your predictions will often be true. And, perhaps, eventually your predictions will be so often true that they become uninteresting, so you must move on to other questions.
Fair enough, I think Kuhn was referring to things like the world-models and you're referring to finding out if the predictions of the model matches reality.
The heliocentric model of the solar system made less accurate predictions than the geocentric model for years, because the geocentric model was mature and had had lots of tweaks applied to it. In that time, you could have asked the heliocentric model to make a prediction, and shown that it was wrong compared to the geocentric model. You would have been wrong to conclude that heliocentrism was wrong though, it just hadn't matured as a theory enough yet.
> Are there things that we can't test that do exist?
Lots of people think so (e.g. unmeasurable things predicted by theory like parallel universes, but also things like evil or God or the color purple), but by definition it's hard to be very sure, or to transfer your own confidence in such things to others.
Lots of these kinds of questions reduce to quibbling about definitons; and also by definition, if we can't test the thing then the universe isn't going to punish us either way for believing or not.
> if we can't test the thing then the universe isn't going to punish us either way for believing or not.
If we can’t test the thing then what we are discussing is faith, not science.
Nothing wrong with faith and beliefs but I think it’s important to differentiate between these things and science because often times science is used as a basis for untestable beliefs and then people really start to think that those untestable beliefs are actually backed by scientific research.
