Non-standard analysis - infinitesimals and archimedean property

I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero.

However, at the same time, they satisfy all the properties of real numbers - so for example, let's call such an infinitesimal $\epsilon$. Then $2 \epsilon$ and $3 \epsilon$ are greater than $\epsilon$. And most importantly, if they satisfy the rules known from real numbers, then they should also satisfy the archimedean property that states that it's always possible to give a number closer to zero than given number.

I've heard non-standard analysis simplifies some proofs. Erm, how can we prove theorems about real numbers in a system that includes objects that don't satisfy the properties of real numbers (infinitesimals in this case, because they are defined to be the number nearest to zero, which in fact doesn't exist)?

• The definition of infinitesimals in non-standard analysis is quite technical (and there is more than one approach), but it qualitatively results in something like what you have sketched. If you want a more precise understanding and/or to clear up some of the points you seem confused about, it would help to know how much mathematics you've learned already. – hardmath Sep 9 '15 at 10:59
• Well, for any infinitesimal $\epsilon$, the infinitesimal $\epsilon/2$ is closer to $0$... – Arthur Sep 9 '15 at 10:59
• If you do do the non-standard analysis properly the infinitesimals actually do exist. Some of their properties has to differ from the properties of real numbers - or they would be real numbers and add nothing to the standard analysis. – skyking Sep 9 '15 at 12:07

An infinitesimal is not "the number that is closest to zero". It is "a number which is closer to zero than any standard real number". You are right that in NSA, every number has a whole "cloud" of other numbers infinitely close to it, not just one.

The preceding remark doesn't depend on the NSA framework we use. In the rest of this answer, everything is about the Robinson NSA framework exclusively. This works in a system of numbers called the hyperreals, which includes the real numbers as "standard" numbers as well as additional "nonstandard" numbers.

The hyperreals and reals share properties through the transfer principle. But you must be careful in using it. For example, the hyperreals are not Archimedean in the sense that any positive hyperreal is larger than $1/n$ for some standard natural number $n$. Yet they are Archimedean in the sense that any positive hyperreal is larger than $1/n$ for some hypernatural number $n$ (which could be infinite). The transfer principle says every property of the reals is transferred to the hyperreals in this way, but that you must replace all of your original standard objects (like the natural numbers) with their nonstandard analogues (like the hypernatural numbers).

Going in the opposite direction, with the right reformulations of classical definitions, the transfer principle lets you prove a theorem in the hyperreals and get the corresponding statement in the reals for free. In a lot of cases these hyperreal definitions are quite intuitive. For instance a function $f$ is continuous if $f(x+h)-f(x)$ is infinitesimal whenever $h$ is infinitesimal and $x$ is a standard real.

I could make some vague comments about model theory, but I think that is best left to someone with more expertise.

• @AsafKaragila Maybe you can make a remark about model theory here. – Ian Sep 9 '15 at 11:18
• I would hesitate to launch into a discussion of non-principal ultrafilters without knowing more about the OP's math background. You could also link to Wikipedia's article Non-standard analysis. – hardmath Sep 9 '15 at 11:37
• @hardmath I don't want to get into specifics, I agree. I just wanted to try to give another vague view on what the transfer principle really says: namely, it's not so much that you have to "replace the familiar objects with nonstandard analogues" but rather "within the model, the familiar objects are the nonstandard analogues, and we can only work within the model". But I am not well-equipped to elaborate further on this point. – Ian Sep 9 '15 at 11:46