This may seem like a naive/stupid philosophical question so I am prepared to get destroyed here but what is the real world relevance of the uncountability of the real number system? I understand that real analysis and as a result all sorts of applied mathematics rely on this property of the real line (and this may be the answer to my question), but can anything in the physical universe actually be uncountable? Or is this a concept that humans needed to create in order to rigorously develop the mathematics needed to measure/study things quantitatively in the real world? Thanks.
2026-03-30 17:14:11.1774890851
Real world relevance of uncountability of R
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I think you have answered your own question: the real numbers are
Note that the rational numbers suffice for recording all measurements.
I would replace "study" by "model". The invented mathematics of the mathematical continuum - calculus and its descendants - turns out to be extraordinarily useful at predicting the (rational) results of experiments.
That success says nothing about what the real world "is" in a philosophical or mathematical sense. It might be a large discrete object. We might be living in some kind of simulation ...