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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.

Nikolaj-K
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    There is the second-order characterization as as complete ordered field. – André Nicolas Mar 24 '13 at 14:33
  • See http://en.wikipedia.org/wiki/Real_number#Axiomatic_approach – lhf Mar 24 '13 at 14:39
  • @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. – Nikolaj-K Mar 24 '13 at 14:40
  • @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. – Nikolaj-K Mar 24 '13 at 14:42
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    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. – hardmath Mar 24 '13 at 14:42
  • @NickKidman: In the usual minimal inductive subset way. If we try to avoid second-order (and there is good reason to do so) one is stuck with the already complete theory of algebraically closed fields of characteristic $0$, within which one cannot define the natural numbers. – André Nicolas Mar 24 '13 at 14:49
  • @AndréNicolas: Ah okay, I seen now that first defining a field with division and then adding stuff making it a non-field-anymore will be difficult. – Nikolaj-K Mar 24 '13 at 14:56
  • @NickKidman: It can be done. To the usual language of fields, add a unary predicate symbol Int, or Nat. You can then add whatever axioms you want, like the full Peano Arithmetic. – André Nicolas Mar 24 '13 at 15:10
  • @AndréNicolas: You're saying you ignore the side that axiomizes the field? Like tha axiom that grants each element a multiplicative inverse. – Nikolaj-K Mar 24 '13 at 15:36
  • @NickKidman, just single out the subring generated by 1. – vonbrand Mar 24 '13 at 15:36

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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.

masmoudihoussem
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    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. – vonbrand Mar 24 '13 at 15:46
  • Can you explain, including definitions and proofs, "C as the rupture body of polynomes in R"? – nilo de roock May 09 '22 at 10:48