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Real numbers without set theory 
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#1
Feb1414, 11:30 AM

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I understand the definition of real numbers in set theory. We define the term "Dedekindcomplete ordered field" and prove that all Dedekindcomplete ordered fields are isomorphic. Then it makes sense to say that any of them can be thought of as "the" set of real numbers. We can prove that a Dedekindcomplete ordered field exists by explicitly constructing one from the ordered field of of rational numbers.
But how do you state the axioms of the real numbers without using set theory? How did mathematicians do it before every useful branch of mathematics was shown to fit inside a set theory? The completeness axiom is the only one that's causing me any trouble: In a settheoretic approach, the axiom can be stated like this: "Every nonempty subset of ##\mathbb R## that has an upper bound has a least upper bound". Is there a good way to state it without reference to sets? I thought about replacing this axiom with one about metric space completeness instead, but the definitions of "metric space" and "limit" mention real numbers, so this sounds circular. The reason I'm interested in this is that it's relevant to my understanding of what mathematics is. ZFC set theory is much prettier of course, but it doesn't seem wrong to define a much smaller branch where the primitives (undefined concepts) are real numbers, addition and multiplication (and perhaps a few more things), instead of sets and set membership. 


#2
Feb1414, 11:51 AM

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#3
Feb1414, 12:05 PM

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Edit: Of course there is always the issue that topologies bring us right back into set theory, but since all we care about is convergence, it might still be possible to avoid sets. Alternatively the metric notion of convergence probably works just fine here. Replace the "real numbers" in the usual definition with your special field and require all the usual metric properties. So long as you can recover the least upper bound property this will check out fine. 


#4
Feb1414, 12:15 PM

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Real numbers without set theory
The existence of irrational numbers goes back to the ancient Greeks, and probably earlier, but their axiom systems (e.g. for geometry) were very incomplete. Maybe Simon Stevin should get the credit, as the "inventor" of decimal notation (published in 1589). But it's worth remembering that Stevin was the first European to publish the general formula for solving quadratic equations, as a calibration point on the history of math. Descartes used the term "real numbers" to mean "real roots of polynomials", which is of course a different notion. 


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Feb1414, 07:23 PM

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I don't want to construct the real numbers from the rationals or anything else. I just want to write down a set of axioms that describe properties of the real numbers (and addition and multiplication) without explaining what real numbers are, just like the ZFC axioms describe properties of sets without explaining what sets are. In this approach, we would define integers and rational numbers as real numbers with special properties. 


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Feb1414, 08:01 PM

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Not to mention the development of calculus! 


#7
Feb1414, 11:39 PM

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They were that late? Shows how little I know about math history. I would have guessed something like 1800.
Anyway, I think jgens is on the right track. I would have to use the axioms he's suggesting, or something very similar. This is annoying, because it seems to require me to write down axioms for natural numbers at least, before I write down axioms for real numbers. I didn't realize that I would have to do that. So I think I will abandon this idea. If I ever need to show someone an example of a nice set of axioms that defines a branch of mathematics other than set theory, I will use Peano's axioms instead. 


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