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Some authors define $\mathbb{R}$ axiomatically. That is, they assume there exists a set $\mathbb{R}$ with binary operations $\cdot$ and $+$ such that the field axioms are satisfied, an order $\leq$ satisfying various axioms and the least upper bound property.

My question is, simply, is such an approach rigorous? The existence of this set is taken for granted!

I have read through some of the answers to this question and the answers to this suggest to me that the only rigorous (from the set theoretic point of view) way to develop $\mathbb{R}$ is constructively (that is, defining $\mathbb{N}$, then $\mathbb{Z}$, then $\mathbb{Q}$ and then using dedekind cuts/cauchy sequences to define $\mathbb{R}$).

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MathematicsStudent1122
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    That is correct, of course. Most people don't bother because proving rigorously that there is a set satisfying the axioms and that it is unique up to isomorphism is fairly laborious. Also at the stage that the reals are introduced most students are not ready for that kind of rigour. – almagest May 08 '16 at 05:43
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    This is what you're looking for – Moya May 08 '16 at 05:43
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    In my way of thought, it is rigorous. The problem is consistency. It could very well be that the existence of such a set would lead to a contradiction somewhere, so building it up from the other axioms instead of postulating another axiom is more to the safe side, consistency-wise (this is just a naive POV). – Aloizio Macedo May 08 '16 at 05:44
  • The approach is rigorous, but in my opinion it looks bit dishonest. Postulating a very deep property of completeness (which is the basis of almost any serious theorem of analysis) as an axiom does not look very satisfying. Moreover we have a discontinuity in the study of number systems. When we study rational numbers we define them in terms of integers and we should similarly define reals in terms of rationals. I mean I don't see such axiomatic approach being used for other systems like $\mathbb{Z}, \mathbb{Q}, \mathbb{C}$. – Paramanand Singh May 08 '16 at 09:17
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    It depends on what you mean by rigorous. On one hand, it's obvious that any theorem you prove from these axioms does in fact hold for any model satisfying them. On the other, it doesn't claim that such a model exists or is unique. However, you could levy the exact same criticisms against set theory or any axiomatic system - one can't exactly side-step the issue of being unable to know if a system is consistent or not. – Milo Brandt May 12 '16 at 02:26
  • The following discussion of what they call "abstraction barriers" in Abelson and Sussman's "Structure and Interpretation of Computer Programs" may be of interest: https://mitpress.mit.edu/sicp/full-text/book/book-Z-H-14.html#%_sec_2.1.2 – Justthisguy May 18 '16 at 20:16
  • You can think of the axioms for real numbers as an abstraction barrier: once you show that an ordered field with the least upper bound property exists, and that there is a unique order preserving field isomorphism between any two of them, there is no need to remember exactly how you constructed it in the first place. – Justthisguy May 18 '16 at 20:18

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Well, the axiomatic approach "works" but only a posteriori, not a priori.

The point is that one can prove the following two claims:

1) There exists an ordered field which satisfies the Least Upper Bound Property (such a field is called Dedekind-complete)

2) Any two such fields are isomorphic (as ordered fields)

Note that proving claim 1 amounts to come up with some construction based on "earlier constructed" objects based on set theory (the natural numbers, then the rationals, etc..)

These two assertions justify the use of the axiomatic approach; In particular, since any two models will be isomorphic, there is "nothing more" beyond the axioms, from the prespective of the theory of ordered fields.

Putting it differently, any claim which can be phrased using $\cdot,+,-0,1,<$ which holds in one model will be true in every model.

Of course, two different models do not need to be identical as sets (i.e the details of the construction can be different), but the differences won't have any meaning from the perspective that interest us.

This last point is quite important, and in fact occurs many times in mathematics; For example, in his book "Naive Set Theory" , Paul Halmos mentions that the exact constructionof the ordered pair $(x,y)$ doe snot matter at all, what matters are the properties we want it to satisy. In some sense, we do not care how exactly it is constructed, since the differences won't affect any aspects which interest us.

Other examples of this phenomenon arise in algebra, for example in the concept of tensor products. One usually states an "axiomatic definition" in the form a universal property, and then proves that:

1) Any two objects satisfying this property are isomorphic

2) There exists an object with the required property

After the two claims were proven, one can safely forget the details of the actual construction (i.e the proof of the second claim)

Asaf Shachar
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