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.
Best Answer
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)