What does existence of the Real numbers mean? 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:


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*Is this observation valid?

*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?)

*How do these axioms help in logical treatments of Natural and Real numbers (what would happen without them)?

 A: I'm going to consider three questions:
What does the existence of the real numbers is needed?


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*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.
