Definition talk:Real Number/Real Number Line

The question is:

"Okay, this gives a list of proofs, but what exactly is this "real number line" defined as?"

It's just a list of proofs justifying the identification of the real number line with the geometrical entity. Feel free to put it into a format that suits our site better - I'm not sure how best to structure it. --prime mover (talk) 09:18, 3 November 2012 (UTC)

I was always taught to call it the Real Number System. It pleased me that this coincided with Definition:Algebraic System. I have recently become aware of the many proofs with "induces" in their title too. (I believe according to LF this would just be a shift from one point to another in the gigantic landscape of Magmas of Sets).
I would like to know PM when you think the verb "induces" is appropriate in a title (the power set is involved a lot isn't it?). I could take \mathbb{R} and equip it with the discrete topology and discrete metric but that's not really an interesting result because you can do it with any.
Btw, on a completely pointless note some of the titles capitalise induce others don't. --Jshflynn (talk) 11:01, 3 November 2012 (UTC)
The trouble with "real number system" is that "system" is one of those lazy but clever-sounding words which is frequently used instead of the specific word that is really required. A "system", as I was taught in one of my courses in higher education, is "anything" made out of bits which "interact somehow". That is of course completely general, and "general" here just means "vague". Thus I would generally discourage the word "system" unless there is a specific meaning to be exploited. As such, the term Definition:Algebraic System is almost as vague - and in this particular context that is too vague to be of any use.
Let me try to explain. The real numbers under addition form an Definition:Abelian Group; under addition and multiplication form a Definition:Field (Abstract Algebra); with the Definition:Positivity Property added on top of that they form an Definition:Ordered Field (a.k.a. totally ordered field but that's a tautology); with the conventional metric forms a Definition:Metric Space. My view is that each of these properties is relevant at a particular level of abstraction: sometimes you need the group properties, sometimes the field properties, and (in the context of topology and metric spaces, which we're currently tightening up) sometimes you need its metric space properties. In this particular instance, the intention of "real number line" is to exploit its ability to be treated as a mathematical model of an infinite straight line in geometrical space - for which purpose it needs to be considered as a Definition:Euclidean Space, that is, as a Definition:Metric Space.
Yes, the Real Number line is an Definition:Algebraic System, but it's a close call, because the Definition:Distance Function is only a "finitary operation on $\R$" by default - a Definition:Metric Space whose domain is not the Real Number Line is (technically) not an algebraic system because $d: M \times M \to \R$ is not an operation (finitary or otherwise) on $M$, unless $M$ is itself $\R$. --prime mover (talk) 11:48, 3 November 2012 (UTC)
Oh, and I'm not discussing the word "induce" here for obvious reasons. --prime mover (talk) 11:50, 3 November 2012 (UTC)
Firstly, I am in complete agreement with you about the overused word "system". Secondly, I understand now that as you equip a set with these relations and operations that others are not mentioned because they are irrelevant in context.
Now on the topic in question I would like to raise a point that may or may not change your mind completely. Looking at Richard S. Millman, George D. Parker: Geometry: A metric approach with models (1990) there is a set theoretic definition of a line:
Let $S$ be a set and $L$ be a set of subsets of $S$ such that $\forall l \in L: |l|>1$ and $\forall x, y \in S: \exists l \in L: x,y \in l$ ($S$ is the set of points and the elements of $L$ are called lines).
In Cantor-Dedekind Hypothesis, Euclid's postulates and the non-mathematical words "right" and "left" are used. So it may be worth taking a different approach (or at least mentioning this different approach). --Jshflynn (talk) 00:11, 5 November 2012 (UTC)
Why should that change my mind about use of the word "system"? --prime mover (talk) 06:09, 5 November 2012 (UTC)
Pardon my ambiguity. The topic in question (that I thought I might change your mind about) was the geometrical interpretation of the real number line.
• The book I mentioned bases geometry on the theory of metric spaces. Lines, triangles, betweeness and other concepts are defined in it in a much more powerful way. The set of real numbers under this framework can be proved to be a line.
• In Definition:Axiom it is stated that one of the ultimate goals of PW is to base mathematics on a handful of axioms and by taking this approach you can have geometry be an extension of set theory.
• While I think the definitions of Euclid should be preserved I think this site should not define line in the way given as mathematics is more powerful than that now.

If you disagree I will not mind. Excessive amounts of studying formal languages have made me distrust geometry to the point of impracticality. --Jshflynn (talk) 10:03, 5 November 2012 (UTC)

I'm sorry, you've completely lost me. I have studied the above paragraph to which you responded with something about me changing my mind, and I'm at a loss to work out what you are specifically referring to. --prime mover (talk) 13:47, 5 November 2012 (UTC)
No worries. I will put up a demonstration of it so you can tell me whether or not it goes against ProofWiki.
Turns out I have an ebook version (unspeakably violated by the reader who scanned his copy with underlining, circling etc. etc.) and I read the first few paragraphs. It seems to be an interesting approach to geometry. However, I don't really like geometry (my intuition for it sucks as soon as the plane is left) so I'll leave others to maybe cover its content on PW some day. --Lord_Farin (talk) 10:40, 5 November 2012 (UTC)