Is 0 an element of the natural numbers?

The post here is relevant:
https://math.stackexchange.com/questions/1220760/what-is-the-difference-between-natural-numbers-and-positive-integers/1220769#1220769

if 0 was natural it wouldn't have taken centuries to even consider it a number

>Is 0 an element of the natural numbers?
If you speak of the set of natural numbers, then it makes no difference. The sets {0,1,2,3,4,5...} and {5,8,9,10,11,12,...} are isomorphic (which for sets just means in bijection).
If you consider the former set with addition, +, then the resulting structure becomes a monoid, a structure with binary operation and unit or neutral element:
0+a=a+0=a.

Whether we have 0 part of the set denoted by N is mere convention

>>8968723
No by definition.

>>8968875
You can create a bijection of all integers from natural numbers. That does not make negative numbers natural numbers.

Namely the pattern 0, 1, -1, 2, -2 etc

>>8968875
what a useless answer

>>8968723
if you want

>>8968723
One professor ones said 0 was in the natural numbers.

Then another professor got really fucking mad when one of the students said 0 was in the real numbers.

>>8968880

The "definition" which you are so sure of does not have consensus. It regularly happens in mathematical discourse that one speaks of "the natural numbers" as being exactly the strictly positive integers. And it doesn't even matter if you try to say that that discourse is "low, undergrad, naive" and so on, or appeal to some particular authority, because this is immaterial to whether there is broad consensus on the matter of convention (because that's all that it really is). And there isn't.

Here's what's true and sensible. It regularly happens that we wish to speak only of the strictly positive integers, for whatever reason. And it also regularly happens that we wish to append only zero to these. Thus, it doens't matter what we call each set-what matters is that we should, once and for all, have an unambiguous name for these two unequal sets.

If I were in charge, I would suggest a completely reasonable convention which was taught to me in high school: the strictly positive integers are the naturals, while their union with zero gives "the whole numbers". Or we could go vice verse for all I care, or use totally different names. What's important is that two different names, whatever they be, be used for the two different sets, and be agreed upon, finally.

>>8968884
I was explicitly giving examples of sets and not saying "the natural numbers" because that's weakly defined.
The infinite set in ZFC generated by e.g.
{}
and
S(x) = x union {x}
has a concise model for + and in that sense one natural "set of natural numbers" has 0. Counting numbers will usually start at 1 (only computer scientist accept being one off the whole time, e.g. speaking of the 0'th entry, 1'th entry, 2'nd entry, for a list of three elements).
All infinite but countable sets are isomorphic (e.g. in a cateogry of sets), but if you consider sets with structures, there are much fewer isomorphisms.

>>8968889
How so.
OP's answer doesn't have "yes" or "no" as answer. What do you want to add that makes it better, I'm very open to critizism.

No

Anyone who tells you otherwise doesn't know what they are talking about.

>>8968889
What a useless objection. He's right.

>tfw some of our professors defined [math]0[/math] to be natural and some didn't so on every exam you had to ask if [math]0[/math] is natural

Is 0 an imaginary number?

Is 0 a set?

>>8968723
Delete this picture right now.

>>8969620

Yes it is. The imaginary numbers are precisely the numbers bi, where b is any real number. 0i, or 0, fits the bill.

There's been a Sparknotes(or same thing) chart on high-school level math on the market for many years, I see it in gas stations and such regularly. It incorrectly states that the real numbers and the imaginary numbers have no element in common and are consequently disjoint sets; this is false, because 0 belongs to both sets, or axes if you like.

Still, 0 is obviously a fairly special and singular number. In particular, note that you can't use 0(i) as part of a basis to generate the complex numbers.

0 is uncontroversially integral, rational (0/1 in lowest terms, note the GCD=1), real, imaginary, and complex all at once. The only sticking point for people is whether to count it as being natural, or not.

>>8969627

No but {0} is.

>>8969610

Due to this existent ambiguity (which further puts the lie to the claim made by >>8968880 ), any professor worth his salt should, at present, be specifying in the appropriate context. It is easily done with just a few characters, and then editing accordingly: "For the purposes of this exam, N = {0,1,2,3,...}" or otherwise/similar setup language. Happily my professors were fairly careful about this point.

>>8968723
Depends on how you define natural numbers. I have found defining natural numbers without 0 to be more useful though.

>>8970744

Both sets {0,1,2,...} and {1,2,3,...} have their appropriate uses, and so both are deserving of distinct, special notations, whatever they might be.

But the set {1,2,3,...} with its ordering has an especially attractive property, which is both obvious and slightly subtle for anyone who has ever had to be careful to count things properly: /the number n is the nth number, or: the nth largest number/ in the set. An identity from size to ordinality.

Also 0 is its own thing that took a long time for people to get their heads around and is a very special number of itself so it makes sense to "introduce" zero as a separate step.

Perhaps instead of: Nine Zulu Queens Ruled China, we might instead say: Nine White Zulu Queens Ruled China.