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Up until now, we have tried to introduce mathematical models to explain physical phenomenon. But we're still struggling to explain many observations with those models. Universe is supposedly infinite, so possibly consists of infinitely many yet undetected phenomena that can't be explained using current models of GR and OM. There's Godel's incompleteness theorem which states that all mathematical truths can't be proved using any consistent set of axioms. This is a limitation of logical deduction in the world of math.

Do scientists consider the possibility that a supposedly infinite universe would always be 'partially unexplainable', no matter what set of logical rules one chooses to describe it?

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    $\begingroup$ 1. There are currently zero physical phenomena or theories that Gödel’s incompleteness theorem is at all relevant to. There have been many incorrect claims to the contrary, usually by people who know more math than physics, who confuse formal difficulties of particular axiom systems they like with actual difficulties of describing real objects. You might have seen one of these incorrect claims. $\endgroup$
    – knzhou
    Sep 11 '19 at 4:40
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    $\begingroup$ 2. Of course there could be things impossible to understand. Everybody in the field knows this and nobody cares because, after the feeling of deep philosophical wisdom wears off, one realizes that saying this mantra just is... kind of useless. We would like to understand as much as we can. Working towards understanding is the game, not sitting in an armchair and declaring that real workers can’t understand. As of yet, there’s precisely nothing to support the armchair people, either. $\endgroup$
    – knzhou
    Sep 11 '19 at 4:42
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    $\begingroup$ My hunch is that people who worry about this tend not to become physicists. The typical physicist, in my opinion, looks at the universe and marvels at how much of it we have managed to understand in just a few hundred years, rather than worrying that we will never understand all of it. $\endgroup$
    – G. Smith
    Sep 11 '19 at 4:44
  • $\begingroup$ I don’t know any physicist who thinks that something totally different that we have no idea about is going on over the cosmic horizon in the parts of our universe that we can’t see. The assumption is “more of the same”. $\endgroup$
    – G. Smith
    Sep 11 '19 at 4:52
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    $\begingroup$ Cool new stuff might be going on in other universes within a larger multiverse, if a multiverse exists. But we should be able to think about it, in theory. The other universes are, some think, just in different effective vacuum states of whatever the larger multiverse theory is. $\endgroup$
    – G. Smith
    Sep 11 '19 at 4:53
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It could be. For example, there could be some (as-yet undiscovered) property of the Universe that is not computable by a Turing machine (but also not useable by us to construct any more powerful computer) to arbitrarily good approximation, in which case, we might well be out of luck when it comes to describing it with mathematics, for the most part. This is not guaranteed, because non-computable doesn't necessarily mean non-describable (c.f. Chaitin's constant), but it could also be that, too, and in any case its non-computability may make it frustratingly difficult to analyze.

There is no guarantee that "reality" has to make things easy for, or even accessible to, us. Thinking otherwise is really just a form of conceit, however much layers of philosophizing you may choose to wrap it up in. Anything we can imagine to fit in a gap in our knowledge just might fit there as long as it doesn't evidentially contradict anything we've already established. It is, say, entirely possible that all elementary particles are actually small gnomes that just happen to follow to the same precision that we've verified as being accurate, all the laws we have already devised for such things. Of course, when it comes to investigating things, we have to have some guide as to what ideas will be most likely, promising, or useful for thinking about, hence if this theory doesn't tell us anything new or useful, it's probably not worth it to spend time or money on, but there is a real and important difference between saying something isn't worth it, and calling it impossible. If nothing else, it maintains an adequate sense of intellectual humility, something I find too often to be missing.

But then you may ask "doesn't this lead us to just give up?" For one, if you're going to use that as an argument for how "reality" is, then you are doing an "Appeal to consequences" fallacy (It also has some shades of Pascal's Wager). On the other hand, no - whether or not to give up is not a function of if we can or cannot understand everything or even just everything accessible to us (another important distinction), because we also cannot prove something is never understandable, either. It could be, or it could be that the next model to understand it is just over the hill, or simply extremely complicated (again, no obligation to make things easy for us). Granted, if after a very long time (which I'd put at hundreds, to thousands, of years or more), it never cracks, maybe we can start to put evidentiary confidence behind such an idea, but not before then. The trend has been clear for a while now that each unknown we've run across has since become known, and as someone who's done a quite fair share of arguing with global heating deniers, I would have to say that one should never focus on a relatively brief, seeming pause as disproving a trend. Trending quantities fluctuate - to refute the trend, you have to see a deviation over a scale that is much larger than that of the natural noisiness within the quantity the trend fits.

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