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.

$\begingroup$ 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 $\endgroup$

$\begingroup$ I will have a look at the second. It seems to have explained things in very simple and clear way. $\endgroup$

4 Answers 4

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

Version 1: $\mathbb$ is not bounded above in $\mathbb$.

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

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 $\overline<\mathbb>$ 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 $\mathbb$ 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 $\mathbb^*$ is defined with infinitesimal elements! (of course that's a non Archimedean Field but interesting enough to study)

$\begingroup$ The way you've expressed the two definitions of the Archimedean property, in terms of $\mathbb$, 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. $\endgroup$

$\begingroup$ @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. $\endgroup$

$\begingroup$ Thanks for this explanation, the two versions are exactly what I needed to get just the right kind of understanding. $\endgroup$

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.

$\begingroup$ I don't think you need the reference to $n$ or $\mathbb$ 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. $\endgroup$

$\begingroup$ At least I tried to make it clear that $\mathbb N$ is not assumed to be a subset of $F$. $\endgroup$

$\begingroup$ Sorry, but could you please explain to dimwits like me the difference between $1+1+\dots+1$ (with $n$ terms) and $n$ ? $\endgroup$

$\begingroup$ 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. $\endgroup$

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.

$\begingroup$ When I say $nx$, I am saying $\sum_^n x$. 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 $\mathbb$ in it. Example: Integers mod $r$. $\endgroup$

$\begingroup$ That isn't the issue. The issue is that when you write $\sum_^n x$, 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. $\endgroup$

Euclidean geometry is Archimedean because given two segments, say $\overline$ and $\overline$ , each segment can be extended by its own length until the combined length is greater than that of the other segment.

In terms of numbers, it means that given two numbers greater than zero, say $s$ and $t$ , either can be added to itself repeatedly until the resulting sum is greater than the value of other number.

Jerome Keisler in an article “The Hyperreal Line” has:

DEFINITION. An ordered field is said to be non-Archimedean iff it has at least one positive infinite element, and Archimedean otherwise.

If a field has an infinite member, than no finite member of the same field can be added to itself enough times to exceed that infinite number.

I would suppose that in light of nilpotent infinitesimals which aren't invertible to form infinites, his defininition would need to be ammended to say "positive infinite or infinitesimal," but I am not positive on that.

So an Archimedean field is one where there are no values so large that you can't add enough units to reach it (so no infinite elements), and no values so small that enough of them wont eventually add up to be a unit (so no infinitesimals allowed).

It has been answered well and plenty. The Surreal numbers form a non-Archimedean ordered field which contains all the real numbers as well as infinitesimals, infinites, and lots more.

If you remove the requirement from the Surreals that every left element is less than every right element, there is no longer a complete ordering. This gives the numbers called "games".