Infinitesimals: Number Systems, Calculus, and Formal Foundations

All flashcards

Infinitesimal

A nonzero quantity whose absolute value is smaller than every positive standard real number.

Why can no nonzero real number be infinitesimal?

For any nonzero real number $x$, the positive real number $|x|/2$ is smaller than $|x|$.

Transfer principle

A principle that carries first-order statements about the real numbers to corresponding statements about the hyperreals.

Nilpotent element in the dual numbers

The nonzero element $ε$ satisfying $ε^2=0$.

Little-o of an input change

An error term whose ratio to the size of the input change tends to zero.

Quiz questions

1. What characterizes an Archimedean ordered number system?Answer: Integer multiples of any positive element can exceed any fixed element.
2. What does the transfer principle do in nonstandard analysis?Answer: It carries first-order statements about the reals to corresponding statements about the hyperreals.
3. Which property defines the nilpotent element in the dual numbers?Answer: It is nonzero, but its square equals zero.
4. For an infinitesimal array of random variables, what must happen for each fixed $ε>0$?Answer: The maximum tail probability in each row, $\max_k P(|X_{n,k}|\geq ε)$, tends to zero.
5. For a suitable function $F$, what does integrating against the Dirac delta at zero yield?Answer: $\int_{\mathbb{R}} F(x)\delta(x)\,dx=F(0)$