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I am not very proficient in either mathematics or physics. So my question might be invalid. Consider a physical theory where quantities/observables only take value from rationals vs a physical theory where quantities take value from reals. This would definitely have wide-ranging implications for what the theory accounts for or predicts. From set theory, we know that there exist infinities which are bigger and denser than reals just as reals are more denser than rationals. So what would be the implication for QM or TOE if the maths suggests that certain observables take values from those denser sets?

Does the multiverse idea already accounts for this? I know the Mathematical Universe Hypothesis by Max Tegmark simply implies a equivalence between mathematical structure and physical existence.

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The question of "how big" is the cardinality of the continuum ($2^{\aleph_0}$) is rather tricky in set theory. It is consistent with ZFC that it could be bigger than naïvely expected (if the negation of the continuum hypothesis is true).

Apart from this, there are not (in my knowledge) fields of numbers of cardinality bigger than the continuum (even the hyperreals have such cardinality). And it is a rather unavoidable requirement of any sensible mathematical theory of QM that observables take values in a field of numbers, if else it would be very difficult (probably impossible) to match the mathematical predictions with the observations.

The question whether surreal or hyperreal numbers (that both contain the reals, even if they have the same cardinality) could be useful to provide a more satisfactory theory of QM is maybe more interesting. The mathematical evidences, such as the transfer principle for hyperreal numbers, suggest that probably a QM theory with hyperreal/surreal numbers would have essentially the same predictive power than standard QM as it is formulated, but would probably be more involved, and would have to be developed from scratch.

One may also think about developing a quantum theory in a different mathematical theory, mainly weakening the axiom of choice (that yields some counterintuitive results). For example, in the Solovay model (ZF+DC) every set of reals is Lebesgue measurable and $L^1$ and $L^\infty$ are duals of each other. The lack of AC for sets with large cardinality may however be rather inconvenient, especially since the algebra of observables satisfying the canonical commutation relations is, for example, non separable (and thus probably not much could be proved on it without the full AC). Nonetheless it may be worth to explore such directions, if not for immediate concrete applicability at least for the sake of knowledge.

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    $\begingroup$ +1. I learnt a lot from your answer. There is a duplicate of my question I am sure , but just in passing would your answer also apply to my (naive/simplistic ) idea of how we treat the "different" infinities in renormalistation in basic QED, which is what I am reading. If I have phrased this badly, my apologies, please ignore it, I will ask later. $\endgroup$ – user154420 Jun 29 '17 at 9:46
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    $\begingroup$ @Countto10 I am not sure I completely understand your comment, anyways if you want to elaborate/ask a question I will try to answer it ;-) $\endgroup$ – yuggib Jun 29 '17 at 9:50
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    $\begingroup$ @BenCrowell About hyperreals I exactly said the same thing as you commented, i.e. that using hyperreals would not give predictions different from the ones obtained using real numbers (and I added "probably" because the transfer principle does not apply to every statement about hyperreals), you should read more carefully... Concerning your point of view, it is in my opinion rather questionable and far too simplistic. Making a mathematical model for physical applications is all about predictive power. $\endgroup$ – yuggib Jul 3 '17 at 8:49
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    $\begingroup$ And the best and most accurate prediction that you can make for the area of a circle of radius one (in whatever unit of measure) - no matter how precise (or limited) your measurements are - is that it measures $\pi$. Any other prediction that you make, and that does not use rely on irrational numbers, is simply less accurate, and thus less useful. And the same can be said about many other less trivial examples, of course. $\endgroup$ – yuggib Jul 3 '17 at 8:51
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    $\begingroup$ @BenCrowell Finally, concerning measure theory, I do not agree as well. Knowing that there are or there are not subsets of the reals that cannot be measured is an important information that has consequences. It means e.g. that you are not able, at a fundamental level, to have information about observables in some regions of space (pathological or not, this gives an intrinsic limitation to the predictive power of some physical theories that are founded on measuring sets of the reals, such as classical statistical mechanics or quantum mechanics). $\endgroup$ – yuggib Jul 3 '17 at 8:55

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