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My dad introduced me to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition?

Specifically, my professor did a proof of $1>0$ which was intended not as an example but a necessary theorem. That theorem would perhaps have been of some use if it was the multiplicative identity element is greater than the additive identity element, not having yet determined what elements of the reals they corresponded to. However, we already had axioms stating that one was the multiplicative identity element and zero the additive identity element (which seems to be neccessary).

To my mind, if it is necessary to prove $1>0$, then we can no longer assume properties of the reals which I would have assumed. But what properties of the reals are we assuming? Then I found myself wondering, have we defined addition? Have we defined multiplication? What are we defining formally here and what are we leaving informal (at least for the purposes of real analysis)?

What basic concepts are necessary as primitive notions in real analysis?

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Kazark
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  • It looks like your axiom define "$0$" to mean "the additive identity" and "$1$" to mean "the multiplicative identity" -- if so, the theorem already says what you want it to. – hmakholm left over Monica Mar 13 '12 at 18:30
  • @HenningMakholm Right, I agree...is there a shade of meaning here I am missing about your comment? – Kazark Mar 13 '12 at 18:34
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    Related to http://math.stackexchange.com/questions/61377/looking-for-proofs-of-basic-properties-of-real-numbers – lhf Mar 13 '12 at 18:40
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    Necessary? All that is necessary is the notion of a set and the membership relation $\in$, together with some standard axiomatization of set theory. Of course, in practice this would make real analysis impossible to wade through. I would suspect that your professor is mostly trying to introduce the class to "mathematical rigour", and perhaps demonstrate the existence of a complete ordered field from notions that are somehow more basic. (You will likely prove that all complete ordered fields are isomorphic, making the particulars of the construction irrelevant.) – user642796 Mar 13 '12 at 18:51
  • @ArthurFischer The class is actually over now. I've been pondering this ever since it finished a few weeks ago. It was short course so he did not introduced fields or isomorphism tho I would very much like to study that. Necessary of course is one of those words that gets sticky in math... – Kazark Mar 13 '12 at 18:59

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Your second paragraph looks like you've been fooled into thinking that just because the axioms mention the symbols $0$ and $1$, these symbols are necessarily supposed to stand for the zero and one you know from primary school. It's good you noticed that this doesn't quite make sense here.

What's going on (I'm assuming, based on what you write, that you're in a somewhat typical first-year analysis course) is that that most of the things about arithmetic on real numbers we learn in school are not really proved there. They are just being asserted forcefully by the teacher and students are badgered into accepting them under threat of failing. There are arguably good reasons for this: high school math is supposed to be useful for students who are not going to be mathematicians, and who don't have the inclination to worry about proof details when they can just be told the facts.

However, this is not a satisfactory level of development for real mathematics, so sooner or later you'll need to go back and prove properly that arithmetic on reals makes sense and behaves like you've been indoctrinated to believe it does. For various reasons, this usually happens "later" rather than "sooner". Until then, what you need to do is keep carefully track of what we have assumed about it, such that we can verify that it will all get proved eventually. These assumptions is where the axioms for a complete ordered field come in.

Therefore, assume that in good time we will find that there is some meaningful set, which we will call $\mathbb R$, with some operations $+$ and $\times$ and a relation $<$ defined on it, such that such-and-such axioms about them hold. These are then the primitive notions: addition, multiplication, and ordering. They haven't been defined, but you've decided once and for all on some properties that their eventual definition will have to satisfy. Another "primitive" notion that is assumed as a background theory is some elementary set theory which will allow you to speak about functions and sets of reals, et cetera.

$0$ and $1$ are usually not primitive notions (but this varies somewhat between developments) but defined as the unique elements that react in a certain way to $+$ and $\times$.

At some time in the careful development based on the axioms, you will have have proved that there's a certain subset of $\mathbb R$ that behaves just like the natural numbers under $+$ and $\times$ -- and from point on you're supposed to suddenly remember basic arithmetic on natural numbers from primary school, including decimal notation and algorithms up to long division, and use that freely in proofs and exercises. (Some lesser authors won't state this point directly, but just seem to tire of the formalism after some time and begin tacitly to assume that arithmetic on $\mathbb N$ works the way it does).

(A different reason for basing real analysis on some explicit axioms is that there are various ways to construct the real numbers working from scratch -- e.g., either by Dedekind cuts or by Cauchy sequences. We want to make sure that the results someone gets in analysis doesn't depend on whether his favorite construction of the reals proceeds this way or that. The clearest way to do that is to develop all of analysis from a small set of "neutral" axioms for the reals that one can then prove is statisfied by whichever construction of the reals one likes).

Curiously, it seems to be relatively uncommon to state an explicit set of axioms for natural number arithmetic at this point. Perhaps educators assume that everybody who reaches a university mathematics class will be able to convince themselves that base-10 arithmetic makes sense. Or perhaps it is just that the things you need to remember about elementary arithmetic is hard to wrap up in a nice tidy package like the complete ordered field axioms. One could of course present the Peano axioms, but getting from there to the fact that it's OK to rely on grade-school long division when doing exercises is not trivial.

hmakholm left over Monica
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  • So you only need one ordering operator to get started? I suppose that's because = wouldn't be considered an ordering operator, and you can define the other ordering operator by negating equals and the one already assumed... – Kazark Mar 13 '12 at 19:49
  • @Kazark: Good point. In fact, I was considering $=$ to be a logical primitive here, rather than a specific property of real numbers, which is why I didn't list it explicitly. This is convenient because different axiomatic theories generally need notions of equality that behave largely the same, so we can conserve effort by not acting like each of them is a new concept. But if you want, it is entirely possible to make it an explicit primitive of the theory instead. You would then need a slew of axioms that promise you explicitly that $a=b$ implies $a+c=b+c$ and so forth. – hmakholm left over Monica Mar 13 '12 at 19:55
  • Or perhaps that wasn't in fact your point? If you have $x=y$ and $x<y$, you can consider $x\le y$ to be an abbreviation of $x=y\lor x<y$ and $x>y$ to be an "abbreviation" of $y<x$, and so forth. – hmakholm left over Monica Mar 13 '12 at 19:57
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The primitive notions of real analysis are the axioms of complete ordered field. Of these, the most important is completeness, which can be expressed in various forms. One of the most popular is the existence of suprema for sets that are bounded above. One can argue that a course on real analysis is about the various forms of completeness as I have mentioned elsewhere.

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lhf
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  • Right. However, there are several (equivalent) ways to provide axioms for ordered fields, and Kazark will have to consult his textbook to find out which exact ones are used in his course. – hmakholm left over Monica Mar 13 '12 at 18:26
  • @HenningMakholm, I've edited my answer. Thanks for the nudge. – lhf Mar 13 '12 at 18:30
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    There are more notions than just these (although I have voted +1). For example, in order to reason about a complete ordered field, there have to be axioms that say that particular subsets of the field could be constructed. Without set existence axioms, completeness becomes essentially empty of content. – Carl Mummert Mar 13 '12 at 18:31
  • @CarlMummert, ok, but I don't think the OP is talking about that. – lhf Mar 13 '12 at 18:48
  • This was basically my answer,so I didn't post it because it would just be a duplicate of Ihf's. However,I would add that this characterization gives the impression the completeness property is all that is needed to derive the calculus in it's entirety from the axiomatic construction of R. This is simply not so-the order properties of the real line in tandem with the completeness property are what give the real numbers their unique field structure,particularly in generating the usual topology on R,which is actually what supports most the structure of calculus. – Mathemagician1234 Mar 13 '12 at 21:41
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I don't want to sound dogmatic, but the most primitive notions of real analysis is found in the theory of magnitudes under addition. Here, there is no multiplication, no $0$ or $1$, or even an ordering of the numbers - all those notions are derived. You are looking at the axiomatic abstraction of the set $(\{x \gt 0 \; | \; x \in \mathbb R\},+)$ where $1$ has no special meaning (no multiplication).

You can pick any element in your abstract system of magnitudes and call it $1$ - you've selected a unit of measure. Once you do that can talk about $1 + 1 = 2$ and the magnitude $1/2$, defined by

$\quad \frac{1}{2} + \frac{1}{2} = 1$

and you are on your way. You still don't have multiplication, but logic will get there when you start working with the concept of repeated addition. Multiplication is a derived notion from this axiomatic platform.

The definition of $0$ and negative numbers will be forced on you as a logical necessity to facilitate working with magnitudes. You could also add $\infty$ to a system of magnitudes, and set $\infty + a = \infty$, but you will it more fruitful to add $0$ 'to left of all magnitudes', $0 + a = a$.

We list the axioms in the next section.

Definition: Let $M \ne \emptyset$ be a set with a binary operation $+$ satisfying the following properties:

P-0: The operation $+: M \times M \to M$ is both associative and commutative.

P-1: $\text{For every } x,y,z \in M \text{, if } z + x = z + y \, \text{ then } \, x = y$.

P-2: $\text{For every } x,y,z \in M \text{, if } z = x + y \, \text{ then } \, z \ne x$.

P-3: $\text{For every } x,y \in M \text{, if } x \ne y \, \text{ then } \, [\exists u \; | \, x = y +u] \text{ or } [\exists u \; | \, y = x +u]$.

P-4: $\text{For every } x \in M \; \text{ there exist } \, z \in M \, \text{ such that } \; x = z + z$.

Before stating the last property, for any set $K \subset M$, we define

$\quad \text{UB}(K) = \{z \in M \; | \; \forall k \in K \; \exists u \in M \text{ such that } (z = k + u) \, \text{ or } \, (z = k)\} $

P-5: $\text{For every } X \subset M \text{ such that } \text{UB}(X) \ne \emptyset,\; \exists \, \hat z \in \text{UB}(X) \text{ such that }$
$\quad \quad \forall z \in \text{UB}(X) \; [z = \hat z] \text{ or } [\,\exists u \in M \text{ such that } (z = \hat z + u)\,].$

Then $(M,+)$ is said to be a system of magnitudes.

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