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I am just asking in general if problems arise in physics, astronomy, or biology which require large cardinalities, i.e. beyond the Reals?

MJD
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Douglas Gray
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    @EzTheBoss2 - What is "cont."? The set of continuous functions has the same cardinality as $\mathbb R$. (A continuous function is uniquely determined by its values on the rational numbers, so it's really just $\mathbb R^{\mathbb Q}$.) – mr_e_man May 19 '23 at 03:53
  • https://math.stackexchange.com/questions/477/cardinality-of-set-of-real-continuous-functions – mr_e_man May 19 '23 at 04:10
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    The cardinality of the Riemann integrable functions, of the Lebesgue measurable sets, of the convex subsets of ${\mathbb R}^n$ (for each integer $n$ greater than $1)$ are each greater than that of the reals, but I don't know whether you'd consider this fact about them being required for something you'd consider as "applied". Also, there might be something relevant in Ulam's 1964 paper Combinatorial analysis in infinite sets and some physical theories. – Dave L. Renfro May 19 '23 at 04:56
  • @mr_e_man Sorry, I meant to just say $\mathbb{R}^{\mathbb{R}}$, it is fixed in the original comment. – EzTheBoss 2 May 19 '23 at 12:47
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    $\mathbb{R}^{\mathbb{R}}$ otherwise denoted ${ f:\mathbb{R} \to \mathbb{R}}$has a greater cardinality than the reals and its elements are used all across the sciences. Specifically if you want a more concrete usage of the set I would check out the calculus of variations and the study of functionals (functions of functions). This type of math was used to formulate the Euler-Lagrange Equations which are essential to modern Physics. – EzTheBoss 2 May 19 '23 at 12:50
  • As physics is the science of measurement, and all measurements can only to be taken to a finite precision - even the definition of what you are measuring only exists to a finite precision - it is not strictly necessary to use anything other than rational numbers in physics. However, doing so makes the mathematics incredibly more difficult, so physicists use mathematics of the real and complex numbers, even though it goes far beyond what can be measured. Technically, you don't need infinite sets at all, but realistically, the $\frak c^c$ of function spaces is going to be maximum. – Paul Sinclair May 20 '23 at 13:05

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