Real numbers without set theory

[Fredrik]

I understand the definition of real numbers in set theory. We define the term "Dedekind-complete ordered field" and prove that all Dedekind-complete 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 Dedekind-complete 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 set-theoretic approach, the axiom can be stated like this: "Every non-empty 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.

 

[Stephen Tashi]

But how do you state the axioms of the real numbers without using set theory? .

Not that I know anything about this, but I think you should specify whether the alternative axiom system for the real numbers must establish a relation between the real numbers and other numbers (e.g. the rationals, the integers). Are you asking how to create an axiom system for the reals that establishes the relation between the reals and, say, the rationals, without ever using the notion of subset?

 

[jgens]

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.

There is another difficulty here that metric completeness actually does not characterize the real line uniquely. You can probably fix this as follows: Require the ordered field to be Archimedean and satisfy the Bolzano-Weierstrass property relative to the order topology. You should hopefully be able to recover the least upper bound property from that.

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.

 

[AlephZero]

How did mathematicians do it before every useful branch of mathematics was shown to fit inside a set theory?

They didn't bother about such things, until troublemakers like Cantor started poking at hornets nests.

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.

 

[Fredrik]

Not that I know anything about this, but I think you should specify whether the alternative axiom system for the real numbers must establish a relation between the real numbers and other numbers (e.g. the rationals, the integers). Are you asking how to create an axiom system for the reals that establishes the relation between the reals and, say, the rationals, without ever using the notion of subset?

I was thinking that I should be able to write down axioms for the real numbers in a form that enables me to take them as the definition of a (small) branch of mathematics, the same way that the ZFC axioms define a (much larger) branch of mathematics. If possible, I don't want to make separate assumptions about integers or rationals.

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.

There is another difficulty here that metric completeness actually does not characterize the real line uniquely. You can probably fix this as follows: Require the ordered field to be Archimedean and satisfy the Bolzano-Weierstrass property relative to the order topology. You should hopefully be able to recover the least upper bound property from that.

I didn't think of that. I will give it some thought.

They didn't bother about such things, until troublemakers like Cantor started poking at hornets nests.

Hm, they had axioms for the natural numbers at least. (Peano's axioms). I would have thought that something similar would have been worked out for the real numbers. But then, maybe it never was, and that's the reason why there isn't a name of a person attached to the real number axioms.

 

[AlephZero]

How did mathematicians do it before every useful branch of mathematics was shown to fit inside a set theory?

Hm, they had axioms for the natural numbers at least. (Peano's axioms). I would have thought that something similar would have been worked out for the real numbers.

I guess we are thinking at different timescales of "history" here. Peano's axioms were published in about 1890. That's a few thousand years after the discovery of irrational numbers (attrib. Pythagoras), two hundred years after the first use of the term "transcendental" (Liebnitz), and more than a decade after e and pi had been proved to be transcendental.

Not to mention the development of calculus!

 

[Fredrik]

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.