# What sorts of models might be untestable?

What sort of models might be untestable? For instance,

• the model only concerns events in the past, or
• the model is too ambiguously specified for the experimenter to know what to measure, or
• we do not have the technology to make the necessary measurements, or
• demonstrating the model is false would require an infinitely long time

What additional flaws, perhaps related to time, might also make a model untestable?

• This testability notion seems close to Popperian falsifiability; that may be somewhere to start your search. Jun 7, 2012 at 17:15
• It would also be great if you would tell us a bit more about the context and motivations behind your question. By the way, thanks for sharing your problem with the community, and welcome to Philosophy.SE! Jun 8, 2012 at 0:04
• Testing the model would require violence or otherwise morally detestable actions. Jun 8, 2012 at 15:28
• I have tried to reframe this in a way that's a bit more focused, but please consider developing your concern a little further, and maybe telling us more about your context and motivations to help improve the chance you'll get a great answer! Jun 10, 2012 at 17:11

## 1 Answer

The Turing Halting Machine? It is untestable because it can't exist. There can be no codification of this concept.

A number of things in Mathematics can be this way. For example Cantor "disproved" by "diagnalization" that an infinity in 2 dimensions cannot map onto the set of integers, since that time people have created functions which take two inputs, where each set of coordinates maps into the set of integers. (2x+1)(2**y) for example, can be proven by induction to map onto the set of integers. It might have been nice to have a test back when Cantor "proved" it. Cantor's mistake was saying that a mapping "used up" the integers, and that if he found an infinite mapping in the diagonals, that everything else must be something "more".

Obviously, anything instantiating infinity by other than induction and mapping would take an infinite time to test. But, also there is the case by contradiction. Turing's Halting Problem, to me, suggests that not every predicate can form a set. It is not clear what we mean when we way a computation "halts", because there can be no understanding of a theoretical set of halting computations. Halting seems to be a description of computations where we have arrived at an answer.

But then I think both of these problems are any atomic definition of "not". Infinity is simply "not finite". Additionally, that's all that Cantor proved: there was no maximum integer `N` which could numerate the set `{ 0, 1, ..., N }`. Of course, `0` is what? Not what we're counting. I think Bertrand Russell had a blind-spot for the anti-atomicity of "not" which made his depression upon Goedel's Incompleteness understandable.

Ironically, your quest in understanding is phrased as a "not". We know things we've tested, but the set of all things "not testable" may be an isomorph of Halting. After all another concurrence of a "halted" computation is a concrete answer We know that some computations yield answers in finite steps (because we have them) and some computations...well, not yet.

Are there parameters for all that a given thing is "not"?