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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 refle

by dkarl 15y ago

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.

masterzora 15y ago
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.
- ggchappell 15y ago
  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.

by dkarl 15y ago

masterzora 15y ago

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.

ggchappell 15y ago
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.