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:
- 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)?