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It is a common practice in real analysis text book to show that a complete ordered field exist, this ordered field is then called the Real numbers. What does this existence mean (where does it exist) and why is it needed?

I also want to know the validity of the following observation (I understand the observation is not very rigorous). It seems that the completeness axiom of Real numbers has to do with the construction of the Real numbers and similarly the successor and the induction axioms in Peano axioms have to do with construction of Natural numbers from other intuitive concepts. I have the following questions:

  1. Is this observation valid?
  2. Why would we need this axioms for Natural and real Numbers (the axioms to do with their construction)? (Why do not we have such axioms for rationals, negative numbers or complex numbers?)
  3. How do these axioms help in logical treatments of Natural and Real numbers (what would happen without them)?
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  • $\begingroup$ Naturals are not constructed, they are given. Otherwise there is no way to abstract over successor (how would you index or count applications?). $\endgroup$
    – law-of-fives
    May 16, 2017 at 3:39
  • $\begingroup$ My understanding is that in Peano axioms zero is given the others are generated Peano Axioms . I guess you take the intuition meaning of succession as given. $\endgroup$
    – abk
    May 16, 2017 at 3:44
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    $\begingroup$ You should distinguish specification of a structure and its realisation (or implementation as one would say in computer science). The axioms serve to specify what we want of our structure, so that we can work with it; the natural numbers need a principle of induction to prove things for all of them, and the reals need to be a complete ordered field. The realisation of these structures (within set theory) serve to show the specifications can be met. For integers and rationals, purely algebraic properties (e.g., rationals are the field of fractions of the integers) suffice as specification $\endgroup$
    – Marc van Leeuwen
    May 16, 2017 at 3:51
  • $\begingroup$ @abk $\mathrm{SUCC}$ is a map from $\mathbb{N}$ to $\mathbb{N}$. It gives $\mathbb{N}$ an ordering. In the set theoretical sense functions don't create values they associate values. You have some unspecified set you call $\mathbb{N}$ and $\mathrm{SUCC}$ sends values from that set back into the set. Does that make sense? $\endgroup$
    – law-of-fives
    May 16, 2017 at 3:58
  • $\begingroup$ @law-of-fives Thanks but what I do not understand is if SUCC was just imposing an ordering on $\mathcal{N}$ then there would be no need for induction axiom. I quote from Wkipedia that considering the notion of natural numbers as can be derived from the axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. $\endgroup$
    – abk
    May 16, 2017 at 4:07

2 Answers 2

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I'm going to consider three questions:

What does the existence of the real numbers is needed?

  • Let $r$ be an odd number such that $r^2$ is even...

  • Consider a triangle with four sides...

Any reasoning which starts with the sentences above is nonsense because such a number and such a triangle don't exist. Analogously, if $\mathbb{R}$ (defined as an complete ordered field) did not exist (i.e. if there were no structure satisfying the axioms of complete ordered field) then the analysis course would be nonsense. This is why the proof of the existence is needed, to show that we are indeed doing something instead of nothing.

What does existence of the real numbers mean?

Real Analysis can be viewed as an axiomatic theory. In this context, the proof of the existence of $\mathbb{R}$ (i.e. the construction of a complete ordered field) means that there is a model for the axioms of a complete ordered field and thus the theory (i.e. the real analysis) is consistent.

Why would we need the completeness axiom?

Without the completeness axiom we cannot do analysis. Take, for example, $\mathbb Q$. It satisfies all axioms that defines a complete ordered field except the completeness axiom and nevertheless is inadequate for analysis.

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    $\begingroup$ Couldn't have said it any better Pedro. Definite (+1) from me. $\endgroup$
    – John W. Smith
    May 16, 2017 at 6:07
  • $\begingroup$ Thank you very much. Your detailed answer does clear things a bit, but if existence of Reals is shown, so we can prove things in real analysis then wouldn't you need to show the existence of any other structure or anything for that matter before proving things about them. For example wouldn't you need to show the existence of sets for set theory or even existence of a triangle in Euclidean geometry? If completeness is needed for proof of things we need to do in real analysis would we ever encounter a thing which requires some other axioms to be added to the list of axioms of real numbers? $\endgroup$
    – abk
    May 16, 2017 at 6:21
  • $\begingroup$ @abk In any axiomatic theory, what you should proof before starting to deduce theorems is the consistency of the axioms, which is done by constructing a model. The existence of the objects of the theory (which are defined in terms of the primitive notions) is a consequence of the existence of the model (which is constituted by the primitive notions). Here I'm referring to the theories based on the set theory (but not to the set theory itself). $\endgroup$
    – Pedro
    May 16, 2017 at 7:38
  • $\begingroup$ For example, in the Euclidean geometry the existence of triangles (defined in terms of points and segments) is a consequence of the existence of a model for the axioms of Euclidean geometry (which contains points and segments). However, sometimes these things are not proved in the "basic" courses. An example where the issue of consistence of the geometries (Euclidean and non-Euclidean) is considered is here. $\endgroup$
    – Pedro
    May 16, 2017 at 7:38
  • $\begingroup$ It is worth to notice that in this (rigorous) exposition of the geometry, the consistency of $\mathbb{R}$ is assumed. With respect to your other example, about the "existence of sets", the things are more complicated, I don't know how the consistency of the set theory is studied and thus I'm not able to talk about it. Your last question is very good, but again I do not have an answer. $\endgroup$
    – Pedro
    May 16, 2017 at 7:38
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Existence really just means consistency with whatever set of assumptions and rules. In the system of Peano axioms, we say the successor of zero "exists" because the Peano axioms and accepted rules of logic lead to that conclusion. We say the predecessor of zero does not exist (as a natural number) since the axioms and rules do not lead to its existence.

Now maybe some will argue about other interpretations of the concept of "existence" but, formally, it is really tied to consistency (as already mentioned in other answers) with whatever assumptions and rules of inference. The term is sort of unfortunate. It is intuitively natural to think it implies the mathematical objects are "out there" floating in the aether. And maybe that is the case, but that is another matter that is sufficiently disconnected from the formal math itself. Rather than "x exists" it might have been better for the standard term to be something like "x membership is inferred" or "x infers" or "x axists" or "x is axiomsted."

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  • $\begingroup$ “We say the predecessor of zero does not exist since the axioms and rules do not lead to its existence.” No. We say that because the axioms specifically rule it out. $\endgroup$
    – MJD
    Apr 8 at 21:40
  • $\begingroup$ @MJD specifically ruling something out as existing definitely does not lead to it existing. $\endgroup$
    – jdods
    Apr 9 at 4:40

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