What's your favorite axiomatization of the real numbers?
The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers.
Personally I like Tarski's axiomatization of the real numbers.
1. Symmetry If x < y, then not y < x
2. Denseness If x < z, there exists a y such that x < y and y < z
3. Separation For all X, Y ⊆ R, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.
Or in plain language:
"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
4. Associativity x + (y + z) = (x + z) + y.
5. Closed under addition For all x, y, there exists a z such that x + z = y.
6. Exclusion If x + y < z + w, then x < z or y < w. Axioms for one (primitives: R,
>What's your favorite axiomatization of the real numbers? >The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers. That's not an axiomatization, that's a construction of the real numbers. Axiomatization is what you wrote below.
Juan Mitchell
What this user said, more or less.
You wrote a set of axioms. Also, since you didn't give axioms for multiplication then you don't necessarily have a multiplication in your models (and if a multiplication is given then there's no requirement that it resemble the one we're used to).
The 14 different axioms are given such that any model of the reals is isomorphic to any other. That is to say that we can reliably interpret them as the real numbers, this is not true of your 8 axioms.
Regarding constructions: >These refer to the way in which one can encode a model of the "real" numbers into another axiomatic system such that they satisfy the axioms. >There are two standard approaches for set theory. Both involve encoding "integers" and "rationals" as sets in the same way (with "addition"s and "multiplication"s encoded in set theory as well). >The first approach takes the "rationals" and defines dedekind cuts over the rationals. After defining multiplication and addition between cuts one defines the set of all cuts and shows that it satisfies the axioms for the reals. >The second approach is similar except your real numbers are defined as cauchy sequences (which are again defined as sets). Whenever trying to encode the real numbers into another theory one must note that only a countable subset of the reals can be explicitly described in the theory and so the approach must rely on the remaining reals "existing" only in the models of the theory and not in the theory itself (in general it's only possible to explicitly describe at most a countable number of things, regardless of theory or language).
Andrew Taylor
>Also, since you didn't give axioms for multiplication then you don't necessarily have a multiplication in your models.
Wouldn't that be a positive? An axiomatization should describe the structure with as few rules as possible. If we can do it without introducing new symbols, isn't that just good and we can add definition of multiplication later if we need it. The standard way of constructing from a ordered field also doesn't have derivation, integrals or tan operation and nobody complains.
Christopher Baker
>Wouldn't that be a positive? Depends. I'll give an informal example. Suppose I tell you I'm going to give an axiomatic system for squares and I do it by giving the following axioms (ignore that one of these is redundant, it's not the point I'm trying to make): >All squares have four sides of positive length. >Opposite sides are the same length. >Adjacent sides are the same length.
Now suppose someone else comes along and says they can give a better set of axioms for Squares like so: >All squares have four sides of positive length. >Opposite sides are the same length.
Now it's true that their system is shorter but it doesn't just describe squares. In fact rectangles also satisfy these axioms. In other words, it is a perfectly fine axiomatic system, but it is an axiomatic system for rectangles, not squares. Though I could always use the rectangle system and define a square as "a rectangle satisfying some extra properties" but there should usually be some reason for doing so. I mean, there's nothing really stopping you from replacing lots of axioms with definitions.
>An axiomatization should describe the structure with as few rules as possible. Not necessarily true. Your set of axioms should contain independent axioms (no axiom should be derivable from the rest), that much is true. However, when choosing axioms you should choose ones that capture some intuitive concept and are easier to work with. You're not doing yourself any favors by choosing an axiomatization that is hard to use and remember just because there's less axioms (especially since the axiomatizations ought to be equivalent).
>If we can.... >The standard.... Do you think multiplication is sophisticated addition? Try defining [math](\pi)(e)[/math] in terms of addition.
Eli Bell
>Wouldn't that be a positive? An axiomatization should describe the structure with as few rules as possible But if you don't define multiplication then you haven't defined the structure of the reals as a field. There is no one algebraic structure of the real numbers, rather there are multiple objects in different categories obtained by adding or removing structure from or to the real numbers in some other category.
Hudson Ward
Actually those are axioms for quadrilaterals :)
Joseph Lee
not that user but does this mean you have to write enough of a description to include all the properties of the real numbers, and not just enough to construct them, to be considered THE axiomatization of the real numbers?
Caleb Sanders
The one where they don't exist, because they really don't.
Christian Turner
>maths cares about metaphysical questions like whether concepts like 'existence' really adds the wild onto the 'berger
