In ZF, is there a total order on the set of functions from $\Bbb R$ to $\Bbb R$?
I was searching online for the answer but only found out that the set of all the rational functions is totally ordered.
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3Every set has a total order. In fact, every maximal (with respect to inclusion) order is total, and maximal orders exist because of Zorn's lemma. – Mariano Suárez-Álvarez Oct 31 '22 at 02:46
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Sorry, I should've specified axiomatic system I'm interested in. Does the set of all the real functions has a total order in ZF? – epsilonQM Oct 31 '22 at 02:51
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What do you mean by "real functions"? – Eric Wofsey Oct 31 '22 at 03:02
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All the functions with real number inputs and outputs that have a domain over all the real number line – epsilonQM Oct 31 '22 at 03:03
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1Does this answer your question? Why isn't there a total order of $\cal P(\Bbb R)$? (notice that if $\lvert X\rvert\ge2$, then using indicator functions $\mathcal P(X)$ can be seen as a subset of $X^X$). – Sassatelli Giulio Oct 31 '22 at 07:25
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Also this – Sassatelli Giulio Oct 31 '22 at 07:30