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The example in this answer proves the fact familiar to CS students - that the "big-O" is not a total order. However, most algorithm running times analyzed using big-Oh notation are not expressed in piecewise form like this example. In fact, most algorithms I am familiar with have a running time expressed in terms of polynomials, exponentiations and logs.

Consider the recursively defined class of functions which includes $f(n) = c$ for any constant $c$, $f(n) = n$, and any functions of the form $f + g, f \cdot g, \log(f), \exp(f)$ where $f,g$ are in the class. Does $O$ impose an ordered partition on this class of functions? The functions with the same big-$O$ growth is in the same part.

Here are my thoughts:

Note that specifying $f \cdot g$ is actually redundant, since $f \cdot g = \exp(\log(f) + \log(g))$. Since the functions are inductively defined, perhaps there is an inductive proof.

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    $\begingroup$ $2n = O(n)$ but also $n=O(2n)$... $\endgroup$ – nir shahar Jun 24 '20 at 23:18
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    $\begingroup$ @nir Yeah, you need to replace "total ordering" with "weak ordering" for this question to make sense. Here's a Scott Aaronson proof of a related theorem on O notation classes. I think I might have seen more about this from him somewhere. $\endgroup$ – Aaron Rotenberg Jun 25 '20 at 6:26
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This was shown by Hardy in his monograph Orders of Infinity.

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