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Of course there may be properties of the real numbers not derivable from the axioms (some Gödelian hand-waving here, I have only studied up to multivariable calc. and only dipped my feet in DEs + linear algebra), but is there anything we know to be true that does not come from the axiomatization of the real numbers as a complete ordered field? A yes or no and an idea of what the statement may be would be fine since I am just asking this because I could not find it on the internet, but any explanation or other depth is absolutely welcome!

Here, what I'm asking is similar to this, if it helps: Are the real numbers the unique Dedekind-complete ordered set? but instead of talking about Dedekind cuts (which are way over my head) in terms of abstract algebra. If it helps though.

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You need to specify what does 'complete' mean here. There are many types.

If you mean Dedekind-complete, then yes, real numbers are the unique (up to an isomorphism) Dedekind-complete ordered field.

However, if you mean Cauchy-complete (that is, all Cauchy sequences converge) then no. There are many Cauchy complete ordered fields. For example, the hyperreal numbers.

However, the real numbers are the unique (up to isomorphism) Cauchy complete Archimedean ordered field. Assuming the Archimedean property is essential, as otherwise even simple limits like $\lim \frac{1}{n} = 0$ cannot be proved.

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  • $\begingroup$ If I understand your answer correctly, you are basically saying the reals are unique (is that right?), although you have to specify a little more than just "complete ordered field". How does the fact that the axiomatization is unique imply that we can solve all known to be true problems using this axiomatization though? $\endgroup$
    – Pineapple Fish
    Commented May 29, 2020 at 13:05
  • $\begingroup$ How do you know them to be true? You have a proof, based on some previous results. How do you know those to be true? You have some more previous results. Keep going like this for any result in Real Analysis, and you will find the axioms of real analysis at the end. $\endgroup$
    – Ishan Deo
    Commented May 30, 2020 at 5:37
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    $\begingroup$ Thanks (sorry it took so long to get back). As per Wikipedia, I now understand that there are several ways to introduce the 'completeness axiom' for the real numbers (en.wikipedia.org/wiki/Completeness_of_the_real_numbers). Coincidentally, I was only familiar with the least upper property so thank you for introducing me to the other two (or three?). Have a great day! (also about real analysis being based off of those is helpful) $\endgroup$
    – Pineapple Fish
    Commented Jun 6, 2020 at 15:58

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