non-archimedean in lay terms

Question

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way over my head ( so I apologize for the large gaps in my informal training ).

The concept I keep floundering in is that of non-archimedean fields.

I understand pretty well what fields are - and I do understand the algebraic structure and ordered field concepts in archimedean fields --- it's the "non"-archimedean part I don't Grok. I am having trouble visualizing this. Well, one example of non-archimedians are infinitesimals - not exactly visualizable either (probably a math-geeky pun there).

Can someone please give an example or two of a non-archimedean structure, object, beasty - but in layman's terms ? (Yes, I have read the wiki stuff.)

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[Edit] found these useful after some comments received:

Intuition behind "Non-Archimedean" -- two senses of "non-archimedean".

Example of a complete, non-archimedean ordered field

And this was a good refresher (for me at least) on ultra filters in this context: A layman's motivation for non-standard analysis and generalised limits

Also curious why an editor removed the Field-Theory tag I put on here. Non-Archimedean Fields are not considered part of Field theory ?? If not, then where's the Non-Archimedean Field Theory tag ? :-P

Answer 1

The cheap version is this: rational functions in one variable $x,$ where a function is called "positive" if it is eventually positive as $x$ goes to $+\infty.$ One function is greater than another if the diffference is positive.

In this field, $\frac{1}{x}$ is smaller than any positive real, yet is also positive. Therefore "infinitesimal"

NOTE: I have not read the Wiki stuff. If you wish detail in something intended as a textbook, I suggest Hartshorne's Geometry:Euclid and Beyond.

Answer 2

The lexicographic ordering on the points in the plane $\mathbb{R}^2$ gives a simple example of a non-archimedean ordered group. In the lexicographic ordering, up-and-down movements (i.e., changes in $y$) are insignificant (infinitesimal) compared with left-and-right movements (i.e., changes in $x$).

Answer 3

Since I see you are in systems assessments rather than technical pure math, I would like to propose to your attention Efthemiou's approach to infinitesimals, which may satisfy your request for a layman's definition.

The Atomic Theory of Calculus by Costas Efthimiou

Of course you have heard many times that all objects in this universe are made of molecules which are made, in turn, of atoms. Atoms are the smallest - once thought indivisible units - of matter. It comes as no surprise to you when I say that my watch is made of atoms, my hand is made of atoms, and so on. But it may come as a surprise to you if I say that all mathematics that studies the continuous and smooth changes of the physical world - that is, all math that is based on calculus - is made also of some indivisible units - the math atoms - that are called infinitesimals. And in the same way that atoms have definite rules of behavior that dictate how they may combine to form molecules and objects, infinitesimals also obey rules that dictate how they can be used to derive all mathematics. Mendeleyev discovered the order of the physical universe, while Newton and Leibnitz discovered the order of the mathematical thought. Since mathematics is the language (tool) of science, understanding calculus is essential for understanding science. And you can understand calculus and how it is applied only if you understand its building blocks, the infinitesimals.