In Is zero a number? it is stated that the natural numbers start with the unit. This certainly comes as a shock, given that the second-order Peano axioms (the definition of the structure of N) start with “There exists an element 0”.
First, Mathochist, welcome to the Straight Dope Message Boards, and thanks for providing a link to the column!
Interestingly, it looks like this is a definitional question with different authorities using different definitions of the natural numbers. When I wrote the Staff Report, I used John Olmsted’s classic The Real Number System (1962) which defines N the Natural Numbers as the set of positive integers {1, 2, …}. And I’ve also now checked H.L.Royden’s Real Analysis (1968) which also identifies the natural numbers as the positive integers. Admittedly, my texts are a little old, back from the days when I was doing my math Ph.D. in the early 1970s.
Background: Peano (in 1889 and after) was trying to define a number set, let’s call it the “Natural Numbers.” His axioms are:
(1) That there is a starting point, some “initial number” that is in the set (we’ll come back in a minute)
(2) That there is an order – that is, given a number n in the set, there is a “successor” number n’ that comes next
(3) That successor numbers are distinct – that is, if n and m are distinct (unequal), then n’ and m’ are distinct
(4) That the initial starting point is not a successor to any number in the set
(5) The Induction Axiom: If there is some property such that (a) the initial number has that property, AND (b) if any number n has that property, then its successor number n’ also has that property; THEN the entire number set has that property.
OK, now where it gets interesting is that a quick websearch for Peano’s Axioms shows two different versions of the first axiom. Some list 0 as the initial number for the set, and others list 1. The remaining axioms are identical in all lists, just slightly reworded.
So, I guess it depends on how you want to define things. If you use 0 as the initial number, then you get a different set of natural numbers than if you use 1 as the initial point. In the Staff Report, I made it fairly clear that I was defining “natural numbers” to be equivalent to “counting numbers”, and I don’t think there’s any ambiguity about “counting numbers.”
I’m kind of time pressed at the moment, so I’m not able to resolve this. There are several possibilities:
A. The definition of Natural Numbers used by Peano in 1889 started with 0, and was changed sometime after that but before the 1960 to start with 1
B. The definition of Natural Numbers used in my texts has changed recently (since the 1960s)
C. There is no universally agreed definition of the term “Natural Numbers.”
It seems to me that starting with zero would give you a problem here.
Take n[sup]2[/sup]=n
00 = 0
11 = 1
Axiom 5 would then have it that all integers would follow this rule.
But 2*2 =/= 2
No it wouldn’t. Induction requires that n[sup]2[/sup] = n implies that (n + 1)[sup]2[/sup] = n + 1 for all n. There’s no implication from 1 to 2, so there’s no problem.
Anyway, there are constructions of N that contain 0, and constructions that don’t. The standard construction has 0 = {}, n’ = n U {n} (the successor of n is denoted n’), and addition and multiplication as defined below:
x + 0 = x
x + y’ = (x + y)’
x * 0 = 0
x * y’ = x + (x * y)
But, there’s no reason why you couldn’t take 1 = {}, n’ = n U {n}, and do some alternate definitions:
x + 1 = x’
x + y’ = (x + y)’
x * 1 = x
x * y’ = x + (x * y)
This has all the properties of the naturals without 0, but most people use the first construction–that’s why I called it the standard.
Anyway, as soon as you construct Z from N, you do get 0 no matter which version of N you start with.
ultra, if the most common use of N is now {0,1,2,…} then I will amend the Staff Report, but I’d like a little verification. As I said, when I did my degree back in the late 60s and early 70s, N was the same as the “counting numbers.” Do you know when or why it changed?
My undergraduate days were '90 to '93 and we were told then that some people counted 0 as a natural number and some didn’t.
When I went to high school, just a few years ago (about 2) we learned that N is {1, 2, 3…} and that zero appears in the whole number unit W {0, 1, 2…} as well as some others (real numbers, intergers)
No, I don’t. In my undergraduate days ('98 to '02), I think I only encountered one author who excluded 0 from the natural numbers. For what it’s worth, the first construction I gave is due to von Neumann, and it may well be that his influence finally won out.
I posted this yesterday and it got eaten by starving hampsters. Dang laundry baskets anyway.
I pulled out my copy of the classic Hardy and Wright “An Intro. to the Theory of Numbers” 5th edition, 1983 printing, orig. pub. 1938. It uses only the phrases such as “integers”, “positive integers” etc. No notation for them at all.
By the time I was in college, all undergrad and graduate courses (CS and Math) used N for {0,1,2,…} and Z[sup]+[/sup] for {1,2,3…}.
The use of “counting” and “whole” numbers in el-hi is inconsistent. They are bad terms anyway and should be thrown out.
Wikipedia does note that some people leave 0 out.
Most of the books I’ve seen lately (especially undergraduate algebra texts at various levels) define the natural (= counting) numbers as N = {1, 2, 3, …} and the whole numbers as {0, 1, 2, 3, …}; but I do have at least one book (a graduate [modern/abstract] algebra text) that considers the natural numbers to be {1, 2, 3, …}.
I guess the bottom line is, if you’re going to write about “natural numbers,” just be sure to specify exactly what you mean.
Judging by the translation from the Latin (*) in From Frege to Godel (Harvard, 1967, p94) ed. by Jean van Heijenoort, Peano’s 1889 version defined N as “number (positive integers)” and defined 1 as what you call the “initial number”.
The use of N may just be an imposition on the part of the translator, but I’d be surprised if they’d changed a 0 to that 1.
(*) I’d have guessed Italian. So, as of this evening, now my best guess as to the last significant scientific work written in Latin.
My maths lecturers (Cambridge, UK if anyone cares) have done it both ways. When looking for maths cites online I generally rely on three sites: http://planetmath.org/encyclopedia/NaturalNumber.html , http://mathworld.wolfram.com/NaturalNumber.html , and http://en.wikipedia.org/wiki/Natural_numbers .
The consensus seems to be:
[ul]
[li]Peano’s original formulation included 0.[/li][li]Now, both ways are used. Typically set theorists like {}=0, and number theorests like to exclude 0. Both ways work fine, but have to include ‘except…’ or ‘…and 0’ in different places.[/li][li]Some people use ‘counting number’, ‘whole number’, or something else, to try to remove ambiguity, but not always the same way! (I rely soly on mathworld, here, I haven’t seen these used since school, and can’t remember how they were defined.) I think N0 means N including 0, but it might be the other way round, I can’t remember.[/li][li]I think ‘positive integer’ including 0 and ‘non-negative integer’ not including zero are universal, but I’ve seen people use them wrong.[/li][li]Usually you can tell from context. If it matters I personally try to make it explicit, sometimes using subscript U{0} or {0}, but that’s just me.[/li][/ul]
Yeah, I wish there was a convention, but there doesn’t seem to be. I hope I’ll be corrected…
[QUOTE=Shade]
[li]I think ‘positive integer’ including 0 and ‘non-negative integer’ not including zero are universal, but I’ve seen people use them wrong.[/li][/QUOTE]
Did you mean this the other way round?
:smack: :smack: :smack: Yes. Damn. Thanks.
“0 and 1 are more or less the same thing anyway.”
-my Algebraic Topology prof
<< “0 and 1 are more or less the same thing anyway.” >>
Quite true, of course, for an algebraist, since each is the unit under a group operation (addition and multiplication, respectively.) A topologist however, would probably say that 0 has a homotopy group equal to the integers, while 1 has a homotopy group of 0 (that is, non-technically, you can shrink the symbol 1 to a point, but you can’t shrink the symbol 0 to a point because it has a hole in it.) A donut is different from a pancake, for a topologist.
Having re-read the Staff Report, it clearly defines what I meant by the Natural Numbers, since it shows the sequence to be {1, 2, 3 …} so there is no ambiguity. Since mathematicians are divided (some are multiplied), and there is no consensus, I shall leave the Staff Report as it stands, with thanks for an interesting thread.
And the donut is not different from a coffee cup. Do we really want these people telling us what’s the same?
(for all you topologists who might be offended: ;))
After reading this thread, I’m starting to think that different specialists might have different definitions. The von Neumann ordinals (the first construction I gave) probably are the standard for set theorists and logicians, but analysts and arithmeticists might have a different opinion.
Maybe you could just put in a parenthetical statement that some mathematicians start it with 0?
I have to admit, though, since the whole point is whether to include zero, equating “Positive integers” with “Non-negative integers” is more confusing to me than this distinction over the Naturals.
Since 0 is neither positive (it’s not greater than zero) nor negative (it’s not less than zero), one normally says the Positive Integers are {1, 2, 3 …} and the Negative Integers are { -1, -2, -3…} and thus the non-Negative Integers are {0, 1, 2, 3…}. I didn’t think that was disputed, nor did I see any confusion?
Achernar is probably talking about this paragraph in the staff report:
That doesn’t seem right to me.