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I think to remember that there is way to uniquely characterize the real numbers $\mathbb{R}$ via an axiom set. I wonder if this is possibly without introducing some notion of the natural numbers $\mathbb{N}$ to do it.

I'm thinking of some direct field axiomatization and then a condition which singles out the reals. Because as soon as you've introduced induction it seems "you've lost" because (I think) you need natural numbers for this.

The question arises because I figure there is a way to specify the natural numbers as a subset of the reals by some well formed expression, but I wonder if that would maybe never be necessary because you can't even speak of the reals without already owning the natural numbers.

I realize that there are axiomatics for directly stating a structure vs. constructions from the naturals to the reals via forming equivalence classes to rationals and then doing something like Dedekind cuts to get the reals. My prefered outcome here would be the answer "you need natural numbers before you can speak of the reals" as this would give them a definite preference and some clear hierarchy.

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    $\begingroup$ There is the second-order characterization as as complete ordered field. $\endgroup$
    – André Nicolas
    Commented Mar 24, 2013 at 14:33
  • $\begingroup$ See en.wikipedia.org/wiki/Real_number#Axiomatic_approach $\endgroup$
    – lhf
    Commented Mar 24, 2013 at 14:39
  • $\begingroup$ @lhf: I feel every approach where you speak of the reals as a set in the sense of a sufficiently strong set theory sense aready knows the natural numbers. $\endgroup$
    – Nikolaj-K
    Commented Mar 24, 2013 at 14:40
  • $\begingroup$ @AndréNicolas: Okay, then given $\mathbb{R}$ via the second order logic axioms, can you single out $\mathbb{N}$ from them via addition axioms (restrictions) for these numbers? I really want to produce the naturals from the reals in that order. $\endgroup$
    – Nikolaj-K
    Commented Mar 24, 2013 at 14:42
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    $\begingroup$ If you are thinking of the characterization of the real number $\mathbb{R}$ as a Dedekind-complete ordered field, the issue is not so much avoiding an introduction of natural numbers $\mathbb{N}$ as the difficulty of expressing completeness via a first-order theory. If we have to introduce the language of sets in order to express completeness, we will have dragged in more than enough machinery to construct the natural numbers from constants 1, 1+1, etc. that are required for our ordered field. $\endgroup$
    – hardmath
    Commented Mar 24, 2013 at 14:42

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in a chronological way mathemticians defined N then Z then Q by equivalent relation then R as the topological complementof Q then C as the rupture body of polynomes in R.

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    $\begingroup$ This is quite wrong... they futzed around with $\mathbb{N}$ and $\mathbb{R}$ (the Greeks did so, but didn't know the properties of "lines" described the reals, they initially believed them to be just $\mathbb{Q}$). In the Renaissance $\mathbb{C}$ showed up to solve cubics, and people started taking negatives (i.e., $\mathbb{Z}$) seriously. Euler made imaginaries into bona-fide numbers for analysis, much later the axioms we use to describe the sets were codified, particularly completeness. $\endgroup$
    – vonbrand
    Commented Mar 24, 2013 at 15:46
  • $\begingroup$ Can you explain, including definitions and proofs, "C as the rupture body of polynomes in R"? $\endgroup$
    – nilo de roock
    Commented May 9, 2022 at 10:48

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