Archimedean Property and Real Numbers

I have few confusions:

a) What exactly is Archimedean Property. What does infinitesimal and infinite numbers do not exist in Archimedian ordered fields mean? Are not 0 and infinity such numbers?

b) What are the surreal numbers? Do they have anything to do with extended real numbers? I mean real numbers and positive and negative infinity. Rudin introduces extended real numbers with these two additional numbers. Does it mean in the field of reals, infinite means undefined and in extended, infinite means defined?

Does this mean extended real numbers are not Archimedian ?

Thank You.

-
If you want to learn about the surreals, the classic intro is Knuth, Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. If you want to learn about a similar system called the hyperreals, try Keisler, Elementary Calculus: An Infinitesimal Approach, math.wisc.edu/~keisler/calc.html – Ben Crowell Dec 26 '12 at 18:16
I will have a look at the second. It seems to have explained things in very simple and clear way. – 007resu Dec 26 '12 at 18:29

The Archimedean Property of $\mathbb{R}$ comes into two visually different, but mathematically equivalent versions:

Version 1: $\mathbb{N}$ is not bounded above in $\mathbb{R}$.

This essentialy means that there are no infinite elements in the real line.

Version 2: $$\forall \epsilon>0\ \exists n\in \mathbb{N}:\frac1n<\epsilon$$ This essentially means that there are no infinitesimally small elements in the real line, no matter how small $\epsilon$ gets we will always be able to find an even smaller positive real number of the form $\frac1n$.

Note that $0$ is not infinitesimally small as it is not positive (remember that we take $\epsilon>0$) and $\infty$ doesn't belong in the real line. The extended real line $\overline{\mathbb{R}}$ is in fact not Archimedean, not only because it has infinite elements, but because it is not a field! ($+\infty$ has no inverse element for example).

You may want to note that the Archimedean Property of $\mathbb{R}$ is one of the most important consequences of its completeness (Least Upper Bound Property). In particular, it is essential in proving that $a_n=\frac1n$ converges to $0$, an elementary but fundumental fact.

The notion of Archimedean property can easily be generalised to ordered fields, hence the name Archimedean Fields.

Now, surreal numbers are not exactly $\pm \infty$ and I suggest you read this Wikipedia entry. You might also want to read the Wikipedia page for Non-standard Analysis. In non standard analysis, a field extension $\mathbb{R}^*$ is defined with infinitesimal elements! (of course that's a non Archimedean Field but interesting enough to study)

-
The way you've expressed the two definitions of the Archimedean property, in terms of $\mathbb{N}$, is not so good. The trouble here is that there are nonstandard models of arithmetic, in which we have infinite integers. By expressing a defining property of the reals in terms of the integers, you've left open what properties the integers are supposed to have, i.e., you've defined "Archimedean" for the reals in terms of "Archimedean" for the integers. It's nicer to say there does not exist any real number $x$ such that $x>1$, $x>1+1$, $x>1+1+1$, ... This looks the same, but it isn't. – Ben Crowell Dec 26 '12 at 16:53
@Ben Crowell Thanks for pointing that important fact. Though, it seems it's "normal in literature" since the books I am using both use integers to define Archimedian property for reals, I will note the point you have clarified. Thanks. – 007resu Dec 26 '12 at 18:23

Similar to the other answers. The Archimedean property for an ordered field $F$ states: if $x,y>0$, then there is $n \in \mathbb N$ so that $$x+x+\dots+x \ge y$$ where we have added $n$ terms all equal to $x$.

Consequences: There are no infinite elements $u \in F$, that is, there is no $u$ so that $1+1+\dots+1 \lt u$ (with $n$ terms) for all $n$.

There are no infinitesimal elements $v \in F$, that is, there is no $v$ so that $v>0$ and $v+v+\dots+v < 1$ (with $n$ terms) for all $n$.

There is no real number called $\infty$, so we say the real numbers satisfy the Archimedean property. The "extended real numbers" do not form a field, but may be useful for certain computations in analysis. Instead of saying $\infty$ is defined or undefined maybe it is better to say whether $\infty$ is an element of the set you are talking about.

The surreal numbers No does form an ordered Field, but has infinite and infinitesimal elements, so the Archimedian property fails in No.

-
I don't think you need the reference to $n$ or $\mathbb{N}$ in the first paragraph, which has the disadvantage of making it sound as though you're defining the Archimedean property of the reals in terms of some assumed properties of the integers (including their Archimedean property, which would then need to be defined). Syntax is a finite thing, so in $x+x+\ldots+x\ge y$, the number of $x$'s is automatically finite. – Ben Crowell Dec 26 '12 at 18:11
At least I tried to make it clear that $\mathbb N$ is not assumed to be a subset of $F$. – GEdgar Dec 26 '12 at 18:14
Sorry, but could you please explain to dimwits like me the difference between $1+1+\dots+1$ (with $n$ terms) and $n$ ? – John Bentin Dec 26 '12 at 22:10
OK, one of the axioms for a field says there is an element called $1$. This is not required to be the real number or natural number or rational number $1$, just something called $1$. That is what I meant. In really beginner books, this thing may have a different name for a few pages to avoid confusion. Maybe something like $\mathbb 1$. But even then, we begin using the notation $1$ for it after a while. – GEdgar Dec 26 '12 at 22:26

The Archimedean property states that if $x$ and $y$ are positive numbers, there is some integer $n$ so that $y < nx$. This is a property of the real number field. It can be shown that any Archimedean ordered complete fields is isomorphic to the reals.

In an ordered field in which the Archimedean property does not apply, there are numbers $\epsilon > 0$ so that $n\epsilon$ will not eventually exceed every element in the field. These are the so-called infinitesimals.

The extended real numbers are not a field. The Archimedean property applies specifically to the reals.

Fields with infinitesimals are studied in non-standard analysis. See this Wikipedia article.

-
Same problem as with Nameless's answer. – Ben Crowell Dec 26 '12 at 18:13
When I say $nx$, I am saying $\sum_{k=1}^n x$. Whenever you have a ring with identity, it is idiomatic to say $nx$ instead of writing out this sum. Said ring may not have an isomorphic copy of $\mathbb{N}$ in it. Example: Integers mod $r$. – ncmathsadist Dec 26 '12 at 20:25
That isn't the issue. The issue is that when you write $\sum_{k=1}^n x$, you haven't addressed the question of what number system $k$ and $n$ belong to. To define this number system (the integers) sufficiently, you need to define it as being Archimedean...which means that you then need to define what it means for a number system to be Archimedean. – Ben Crowell Dec 27 '12 at 0:49