Articles of nonstandard analysis

Why Cauchy's definition of infinitesimal is not widely used?

Cauchy defined infinitesimal as a variable or a function tending to zero, or as a null sequence. While I found the definition is not so popular and nearly discarded in math according to the following statement. (1). Infinitesimal entry in Wikipedia: Some older textbooks use the term “infinitesimal” to refer to a variable or a […]

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view the Fourier transform as a projection of a vector from the Hilbert space of square-integrable functions on a certain orthonormal basis. It seems, though, […]

What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing them very related to each other. In one the product is not explicitly defined (it is said that it is the result of a series of cuts) in the other it is stressed that $\epsilon^2=0$ is […]

Why hyperreal numbers are built so complicatedly?

I have seen approaches at building hyperreal systems by using complicated notions like ultrafilters and the like. Why not just postulate the existence of infinitesimal element $\varepsilon$ and infinite $\omega=1/\varepsilon$ like we do with complex numbers and build a field system around them?

Basic question about nonstandard derivative

I’m trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 – \frac{1}{2}x^2}{\epsilon} \right)$ for some infinitesimal $\epsilon$. This works out to $f'(x) = st(x + \frac{\epsilon}{2}) = st(x)$ So the derivative of $\frac{1}{2} x^2$ is the standard […]

Which are the mathematical problems in non-standard analysis? (If any)

I would like to learn non-standard analysis, at least the basics of it. I will make use of this book: Elementary Calculus: An Infinitesimal Approach (Dover Books on Mathematics), by H. Jerome Keisler. Before anything else, please let me take up some links that DO NOT have a fully fleshed out answer to my question, […]

What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes of calculus on a sound footing. From what I have read about hyperreal numbers I understand that they are an extension of […]

What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb R:~~~x + a\cdot \epsilon<y+b\cdot \epsilon<z +c\cdot \epsilon \end{equation} This, together with the ordering of $\Bbb R$ being a subset of the […]

Why adjoining non-Archimedean element doesn't work as calculus foundation?

Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used in non-standard analysis. Why does this approach fail?

l'Hopital's questionable premise?

Historians widely report that l’Hopital’s 1696 book Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ (sometimes written as $y+dy=y$ as in Laugwitz 1997). I used to believe this until I looked in l’Hopital’s book and did not find any such equation. What I […]