Whether N includes 0 or not depends on how I feel and what best serves my purposes at that particular moment, which is why I cannot vote in this poll, but when I’m teaching an analysis class I tend to exclude it, while when I program I tend to include it. To avoid all confusion, if I am to use the natural numbers, I will previously define what I mean by them.
The OP really should have included a third option, “it depends” in his poll.
This is how I was taught, as well. That “Natural” are the “counting” numbers that people “naturally” use, starting with 1.
HOWEVER…my 5 year old starts counting with zero. Weird. No matter if she’s counting “by ones” or “by twos” or fives or tens, she always starts with zero. Must be something they’re teaching her in school now. So I have no idea what they’re going to teach her about the set of Natural numbers when she gets to that point.
A maths wiz in the making..
lol, maybe…or maybe just a parrot. Hard to tell with rote memorization stuff.
I was impressed with a question she asked me about 8 months ago, though. “Mama…does Infinity go backwards, too - *before *zero?”
:eek: Seemed like a pretty advanced concept for a little kid to come up with all on her own! But she claims she “just thought of it” and no one suggested it to her.
And I asked her why she starts at zero and if they taught her that at school and she said no, they didn’t; at school she starts with the first positive integer of whatever she’s asked to do. At home, she starts with zero. “It just makes *sense *to start at zero,” she said, “If it’s “by twos”, then two has to come from zero. But my teacher doesn’t like it.”
So I retract my previous post which implied that they’re teaching kids to start with zero when counting now. Evidently, my kid is just weird.
No it isn’t! 
I was taught (in the US) the Natural numbers, the Whole numbers, and the Integers as above, though I don’t remember if we used N and W to represent them. I remember that a stylized Z represented the integers.
One important aspect of these sets is that they all have the same cardinality, aleph(null), as they can be put in 1..1 correspondence with the natural numbers, like rational numbers but unlike real numbers (potentially disputed).
(emphasis mine)
What? Is there dispute that [; \mathbb{R} ;] has a larger cardinality than [; \mathbb{N} ;], or have I misunderstood you? I actually thought that was fairly obvious.
At least during my grad school days back in the 1980s, the definition of N = {1,2,3,…} was standard. I can’t recall a single grad-level text that defined the naturals to include zero, and that was the accepted definition in every class I took where the naturals needed definition.
I remember that the notation N[sub]0[/sub] was frequently used to represent the union of the naturals and zero.
We’re getting further off topic..and deeper into the dark regions of my brain..areas I haven’t checked into for a long time. Anyway..
I recall N, W, Z, and Q (rationals) were all sized Aleph0.
R (real) was sized Aleph1.
I remember the proofs for the Aleph0 collection. I don’t recall off hand how we proved that R was bigger. Perhaps Jamaika, who clearly has more up-to-date math skills than I do, recalls (or disagrees?)
I kinda miss this stuff..
-D/a
It’s a corollary of Cantor’s Theorem (the cardinality of a set is strictly smaller than the cardinality of its power set), proved by diagonalization.
Interesting how this has gone from almost 50/50 to one-third/two-thirds split. There’s still no clear consensus, but it looks like there might almost be a consensus on the SDMB. The ‘almost’ part interests me.
The natural numbers are whatever is convenient for me to tell my classes
In all seriousness, I make sure to define the natural numbers in the classes I use them in, and then to stay consistent through the semester. I also tell my classes that there is no universal consensus on whether the naturals include 0, so it is important to check your textbook author’s definitions and notation.
Incidentally, notation that I’ve seen seems to favor 0 as a natural number, since then Z_+ or N_+ (using + as a subscript, or occasionally superscript) becomes a convenient and relatively intuitive way to represent the nonnegatives without introducing new notation like W. I’ve also never seen W used in a textbook since high school.
As a personal feeling, I think 0 ought to be included.
Do you mean point-set topologists here? (I’m guessing based on your user name). I haven’t asked around, but I’d be surprised if algebraic topologists leaned in this direction, since 0-cells are so important in unstable constructs like simplicial and CW complexes.
Whoops!
The set of rationals, the set of naturals, and the set of integers are all of the same cardinality (or ‘size’ for those playing at home). By custom, we call this cardinality aleph(null). The reason for the custom is that aleph(null) is the least infinite cardinality, and aleph(one) is the next one, etc.* (a very good question here would be: “why is there a least infinite cardinal, and a second least, etc?” and the answer would have to do with the particular axiomatization of set theory that is most popular, namely ZFC)
The reals has the same cardinality as the power set of the naturals, which, as Capt. Ridley’s Shooting Party points out, Cantor proved has a cardinality strictly larger than aleph(null). The fact that is “in dispute” is whether the set of real numbers has cardinality aleph(one). This question is called the Continuum Hypothesis.
However, it’s not actually in dispute. The truth is much stranger. It turns out that the most popular axiomatization of set theory, ZFC, is not strong enough to prove that the set of reals has cardinality aleph(one)** AND the most popular axiomatization of set theory, ZFC, is not strong enough to prove that the set of reals does not have cardinality aleph(one)***.
*See! We start counting at zero!
**This was proved in the 60’s by Paul Cohen. In the course of proving this, he invented the method of set-theoretic forcing, which proves all kinds of surprising things.
***This was proved in the 30’s by Kurt Godel, who also proved the famous Incompleteness Theorem(s). This states that there is a sentence which is true but not provable! But to prove the bit about cardinalities, he discovered the Constructible Universe, which is a (relatively) concrete model of the universe of sets. The Constructible Universe is also used the show the consistency of the Generalized Continuum Hypothesis, as well as the Axiom of Choice.
What’s yellow and equivalent to the Axiom of Choice?
What’s purple and commutes?
Other than with aleph above, I’m not sure how anyone can start counting with zero. Sure, you can say 0,1,2,3,4,5…, but if you actually count like that, you’ll always think there is one less item than there really is.
An abelian grape.
I was taught the above, probably in high school. Never realized there was controversy over whether the natural numbers included zero.
Zorn’s Lemon