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Comment by stiff | Hacker News Reader

I doubt mathematicians commonly use zero as an index of the first element of a collection. Matrices and vectors which are closest to arrays are indexed from one. In the aleph notation aleph null is different from all the other alephs because it represents the cardinality of an infinite, but _countable_ set, so the notation makes sense and it doesn't have any disadvantages.

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\aleph_0 is the first countable infinity because the alephs (like the beth numbers, and the levels of the cumulative hierarchy) are indexed by ordinals, which start from zero, i.e. the empty set in the usual constructions of von Neumann and Zermelo.

However, whether we start from 0 or 1 is ultimately arbitrary and more a question of symbol choice than semantics, since the sequence of natural numbers beginning with 1 is isomorphic to the sequence of natural numbers beginning with 0. Set-theoretically it makes a little more sense to start with 0, but when considering arithmetic alone, it really makes no difference.

There are many other examples. Topological spaces can be described as T0, T1, T2, etc. to indicate the separation axioms that hold on them. In part this reflects the tendency for there to be a first case that is degenerate or that common sense overlooks; for example, lines are one-dimensional and planes are two-dimensional, but mathematicians start with points, which are zero-dimensional.

It feels like this is related to the question of whether the "natural numbers" includes zero or not. In the set theory classes I took, we used N (in "blackboard bold") for the natural numbers 0, 1, 2, 3, ... and Z+ (in "blackboard bold") for the positive integers. My professor's pragmatic take on the controversy was that Z+ already means the positive integers, so it seemed like a waste to let N mean the same thing as Z+ and have no symbol to denote the nonnegative integers.

I always liked my algebra professor's way of getting to the same conclusion of \mathbb{N} including 0:

The natural numbers are how we count things. How many moons does Earth have? One. How many eggs are in the carton in my fridge? Twelve. How many elephants are in this room? Zero.

Incidentally, all of my professors agreed with this and most of them used 0-indexing unless some insight or ease of work was exposed by a different indexing scheme.

When I started grad. school in math, I hung out with a bunch of set theorists & topologists. And they all used \mathbb{N} to mean {1, 2, 3, ...}. Their reasoning was that we already had a symbol for {0, 1, 2, 3, ...}, namely, \omega, the first infinite ordinal. However, my experience in the years since, suggests that your convention is the more common one.

BTW, there is an important point made in your comment, one I had to hammer on repeatedly when I taught discrete math: that the answer to a "How many ... ?" question is a nonnegative integer, i.e., an element of {0, 1, 2, 3, ...}. I've found that this fact is surprisingly difficult for many people to grasp.

Series, sequences, polynomial coefficient are indexed starting with zero. All of them are rather often used and important.

There are different conventions on this. Some textbooks have sequences starting from one, some from zero.

There's an advantage to starting from zero if you want to work with Taylor expansions.

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