Non-Standard analysis and infinitesimal

Question

Can someone please explain how Non Standard Analysis is used to justify infinitesimals?

I am not very clear about this but apparently it has something to do with hyperreals.

Answer 1

The "infinitesimal", or a quantity which is smaller than any non-zero real number but still non-zero, is a non-rigorous idea that had been a weak link in early calculus, which extensively relied upon it. The Leibniz notation $dy/dx$, for example, is originally supposed to mean "an infinitesimal change in $y$ divided by an infinitesimal change in $x$". Well, it worked, but exactly why eluded the early mathematicians who developed calculus.

The infinitesimal problem was first resolved by looking at limits not as precise evaluations of a function at an infinitesimal value, but a somewhat more complicated definition (the $\epsilon$-$\delta$ definition) that made the concept rigorous and logically sound. However, people have always wondered if it is at all possible to actually extend the real numbers in a way (much like the extension to complex numbers) which makes it satisfy two properties:

1. It has to preserve certain properties of the real numbers familiar to us, and
2. It has to contain elements such that any "infinitesimal number" that arises in early calculus corresponds to a precisely-defined element.

Such an extension is called the hyperreals. It justifies the infinitesimal numbers because they are now well-defined: say one wishes to take the derivative of $x^2$ at $x=3$. The definition gives us

$$\lim_{\Delta x\to 0}\frac{(3+\Delta x)^2-3^2}{\Delta x}$$

Doing some quick algebra works this out to $6+\Delta x$ where $\Delta x$ is infinitesimal. In the standard real numbers this word has no precise meaning: it's simply a figure of saying that we happen to be able to manipulate as if it is zero, but without the restrictions (in the simplification above, for instance, we have to divide the numerator and denominator by $\Delta x$, which we cannot do if it is zero). In the hyperreals, however, $6+\Delta x$ is a specific element which is not equal to $6$. However, an equivalence relation called the standard part on a subset of the hyperreals (it is only defined on a subset because the hyperreals actually contain infinite elements as well) gives a homomorphism from the hyperreals to the reals, which can be considered as the hyperreals modulo this equivalence relation. So taking the standard part of $6+\Delta x$ easily shows that the real derivative is $6$. (In fact, any hyperreal construction necessarily admits infinitely many infinitesimals in the equivalence class of any real number.) This would give us property 2 as we discussed above.

What, then, about property 1? It turns out that the hyperreals can preserve many properties of the reals, but not all of them. The ones that they do preserve are the first-order properties. These are properties that can be expressed as a finite formula using quantifiers on the elements, but not on subsets. For instance, "for any two real numbers there is a real number strictly in between" is first-order, hence it's preserved in the hyperreals. On the other hand, the infinitary statement "for any real number $x$ there exists a natural number $n$ such that $x<\underbrace{1+1+\cdots+1}_n$" is not preserved in the hyperreals. (In particular, the field axioms are first-order, so the hyperreals form a field.) This preservation is called the "transfer theorem", a straightforward consequence of the standard construction of the hyperreals as an ultrapower of $\omega$ copies of $\mathbb{R}$ modulo some suitable ultrafilter; Łoś's theorem shows that the first-order properties are all preserved.