[acmiyaguchi]

A first or second year (undergrad) student doing theoretical science to this level is astounding to me. It was an upper-division course for me, albeit a prerequisite for things like compilers and cryptography, so I'm sure it could be put earlier in the journey.

Sipser's "Introduction to the Theory of Computation" was the book I had to read, and it certainly makes this post more accessible.

[auggierose]

Interesting, Sipser starts the natural numbers with 1 in this book, instead of 0.

I like this choice, and I would like to use it myself. But if ℕ starts with 1, what do I call {0} ∪ ℕ? I guess ℕ₀ is a reasonable choice.

[thaumasiotes]

> I like this choice, and I would like to use it myself.

It's not a great choice, purely for the reason that there's already a convenient name for the positive integers, ℤ⁺.

[auggierose]

I've just checked with a few math books I like, and they are using ℕ and ℕ₀ throughout. What I don't like about ℤ⁺ is that 1) it talks about integers, although negative numbers might not be relevant in the current context at all, and 2) it's not really clear if ℤ⁺ denotes ℕ or ℕ₀. Of course, you have the same ambiguity problem with ℕ.

Oh, see how I numbered this 1) and 2), not 0) and 1)?

[thaumasiotes]

> Oh, see how I numbered this 1) and 2), not 0) and 1)?

So what?

> 2) it's not really clear if ℤ⁺ denotes ℕ or ℕ₀.

On the contrary. ℕ is ambiguous between the positive integers and nonnegative integers, though in my experience it's usually the nonnegative integers.

But ℤ⁺ is absolutely unambiguous. It's the positive integers.

And since you frequently need to refer to the nonnegatives, it makes sense to have a symbol for them too. There's almost no benefit to having ℕ available as a synonym for ℤ⁺. There's a lot of benefit to having it available as a synonym for ℤ\ℤ⁻.

[auggierose]

> So what?

Well, it means I usually need either ℕ or ℤ, not ℕ₀.

> And since you frequently need to refer to the nonnegatives

It depends on the context. You don't use ℕ₀ much, if you agree that counting starts at 1. You then use either ℕ or ℤ, which just looks cleaner. No reason to complicate such a simple concept as {1, 2, 3, ...} with something complicated such as ℤ⁺. As I said before, to understand {1, 2, 3, ...}, you don't need the concept of negative numbers at all, so why drag it in through confusing symbols?

> But ℤ⁺ is absolutely unambiguous. It's the positive integers.

Or the non-negative integers. And that's not just my opinion. Here is what ChatGPT has to say about that:

*system*: You are a mathematics professor, and happy to chat about mathematics and the philosophy of mathematics.

--------------------

*user*: What would you say ℤ⁺ stands for? Are there multiple options?

--------------------

*assistant*: ℤ⁺ typically stands for the set of positive integers. This usually includes all the whole numbers greater than zero. However, there can be some variation in its definition depending on the context.

In some contexts, ℤ⁺ might be defined to include zero, so it would represent the set of non-negative integers. This is less common, but it does occur in some areas of mathematics.

The symbol ℤ comes from the German word "Zahlen," which means "numbers." The superscript "+" is used to denote that we're only considering the positive members of this set (or non-negative, in some cases).

So, yes, there can be multiple options for what ℤ⁺ represents, but they are all closely related. It's always a good idea to clarify these kinds of notational issues when they arise, to avoid any potential confusion.

[tromp]

> if you agree that counting starts at 1

I argue that counting really starts at 0 in https://news.ycombinator.com/item?id=33022031