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Given any two real numbers $x,y$ we can compute $$\max(x,y)=\frac{|\,a-b\,|+a+b}{2}$$ using "nice" functions and operations (adding/substracting, absolute value, halving). Inductively we can calculate $\max(x_1,\dots,x_n)$ for any finite collection $x_1,\dots,x_n$ of real numbers.

Now, given a (non-empty) set $A\subseteq\mathbb{R}$ is there a formula to compute $\sup(A)$ in general?

The first problem that I see is that the existence of suprema is an axiom and thus is maybe "out of reach" for operations defined using every other axiom but this one. This already makes me suspicious of a "nice/elementary" formula/algorithm taking a set and gibing its supremum back. In a possible answer please state why your formula is "elemental".

My first attempt is to use something like $\sup(A)=\Vert \mathbf{1}_A\Vert_\infty$ where $\mathbf{1}_A(x)=x$ is the identity restricted to $A$. I see many problems with this: first, while $$\Vert\mathbf{1}_A\Vert_p = \left(\,\int_A |\,\mathbf{1}_A\,|^p\,\right)^{1/p} $$

is "nice", $\sup(A)=\Vert\mathbf{1}_A\Vert_\infty=\lim_p\Vert\mathbf{1}_A\Vert_p$ is taking the limit and I would be willing to admit limits as "elemental" I feel like there is a "computability gap" (computable meaning "an explicit calculation exists"). Also, the value of the integral does depend on the "endpoints" of $A$ i.e if $A$ is an interval

$$\sup A=\lim_p\left(\,\int_A\Vert\mathbf{1}_A\Vert^p\, \right)^{1/p}=\left(\,\frac{\sup(A)^{p+1}-\inf(A)^{p+1}}{p+1}\,\right)^{1/p}$$

You also have to asume that $A$ is lebesgue measurable and while it may not depend precisely on $\sup$ and $\inf$ it certainly needs a lot of info of $A$. The formula for the $\max$ didn't need anything about the numbers: you have the numbers, you have the maximum, no previous checking.

Also, an integral is kind of like a limit and maybe we will be suspicious.

Sorry for the lengthy question.

Thanks!

augustoperez
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  • It's unclear what your asking for: $\sup(A)$ is a formula for $\sup(A)$! Given a computable representation of a set $A$ of real numbers, there is no algorithm for computing $\sup(A)$. – Rob Arthan May 23 '20 at 22:42
  • @Rob Arthan well yes, and max(a,b) is a formula for the maximum of x and y. I'm looking for a formula that depends on the elements of the set without any assumption on the type of set, the type of numbers etc etc – augustoperez May 23 '20 at 22:52
  • But if you're prepared to allow formulas involving limiting processes like integrals, the usual definition of $\sup$ is logically simpler than the description of the results "calculated" by those processes. Have you heard of the Hilbert $\epsilon$-function? That might help. – Rob Arthan May 24 '20 at 19:28
  • @RobArthan yes I know that the definition of sup is simpler. I only want to know if there is some standard procedure/formula to compute the supremum of A based only in its elements. The way you guess the supremum and compute it usually depends on the set in question. For max(a,b) you don't need to know anything about a or b, only that they are real numbers. The formula 1/2(|a-b|+a+b) is logically more complicated but works everytime and is the same in every situation – augustoperez May 26 '20 at 17:54
  • @RobArthan also I'm not really sure if I'd be satisfied with a limit for answer because the way you compute a limit depends a lot on the expression in question. There is not a method to compute all and every limit – augustoperez May 26 '20 at 17:57

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