Zero is a natural number. It is in the axioms of Peano arithmetic, and any other definition is just teachers choosing a taxonomy that best fits their lesson.
It is, but it need not be. In the category of pointed sets with endofunctor, (Z_{\ge 1}, 1, ++) and (Z_{\ge 0}, 0, ++) are isomorphic (to each other, to (Z_{\ge 937}, 937, ++), and to any number of other absurd models), so either would do equally well as a model of Peano arithmetic.
Yes, agreed, there is other algebraic structure that can tell the difference, but Peano arithmetic by itself cannot.
Despite the name, in the usual mathematical meaning of the term, Peano arithmetic does not define arithmetic at all, only the successor operation, and everything else is built from there. Once we have those, for the model (Z_{\ge 0}, 0, ++), we certainly usually do define x0 = 0 for all x; and, you're right, if for the model (Z_{\ge 1}, 1, ++) we defined x1 = 1 for all x (as no-one could stop us from doing), then we'd just be dealing with "0 by another name." But it might be equally sensible, if our model of Peano arithmetic is (Z_{\ge 1}, 1, ++), to define x1 = x for all x, in which case we recover the expected arithmetic.
2 of the axioms are:
1. For all x, x*0 = 0
2. For all x, y: x*S(y) = x*y + y
(Now having written that and looking back, I see that, in my previous post https://news.ycombinator.com/item?id=43442074, I wrote "Despite the name, in the usual mathematical meaning of the term, Peano arithmetic does not define arithmetic at all, only the successor operation, and everything else is built from there." Perhaps this infelicitious-to-the-point-of-wrong wording of mine is the source of our difference? I meant to say that Peano arithmetic does not axiomatize arithmetic at all, but that arithmetic can be defined from the axioms. Thus the specific definition x[pt] = [pt] is eminently sensible if we consider the distinguished point [pt] to be playing the usual role of 0; but the definition x[pt] = x is also sensible if we consider it to be playing the usual role of 1, and even things like x[pt] = x + x + x + x + x can be tolerated if we think of [pt] as standing for 5, say. The axioms cannot distinguish among these options, because the axioms say nothing about multiplication.)
Enderton, “A Mathematical Introduction to Logic, 2nd Ed.”, p,203,269-270
Kleene, “Mathematical Logic”, p.206
EDIT: It seems like you're talking about Peano's original historical formulation of arithmetic? That's all well and good but it is categorically not what is meant by "Peano Arithmetic" in any modern context. I've provided two citations from pretty far apart in time editions of common logic texts (well, "Mathematical Logic" is a bit of a weird book, but Kleene is certainly an authority) and I hope that demonstrates this.
There's a lot of reasons that the theory is pretty much always discussed as a first-order theory. The biggest, of course, is that when taken as a first-order theory it fits neatly into the proof and statement of Godel's Incompleteness Theorems, but iiuc it's just generally much less useful in a model theoretic context to take it as a second order theory (to the point where I only ever saw this discussed as a historical note, not as a mathematical one).
EDIT 2: This is all a digression anyway. Both first- and second-order PA label the start of the Z-chain as 0; so any model of PA contains 0 when interpreted as a model of PA.
But these are all referring to Peano arithmetic as a model of the theory of the natural numbers. And that seems a bit silly: the impact of Peano's work wasn't because he showed that there was a model of the theory of the natural numbers, which everybody believed if they bothered to think about it, but because he showed that all you needed to make such a model was a successor operation satisfying certain axioms. Yes, they may be less model-theoretically congenial because they're second order, but to change Peano's work from what he did historically and still call it Peano's seems strange to me. (I'm fine with dressing it up in modern language, and calling it an initial object in the category of pointed sets with endofunctor, which perhaps is biased but still seems to me to be capturing the essential idea.)
Certainly I was taught the second-order approach, though it was as an undergraduate; I've never taken a model-theory class. As I say, I'm away from my library and so can't consult any other sources to see if they still teach it this way, and anyway I am a representation theorist rather than a logician; but, if the common logical approach these days really is to discard Peano's historical theory and to call by Peano's name something that isn't his work, even if it is more convenient to use, then I think that's a shame from the point of view of appreciating the novelty and ingenuity of his ideas. But just because I think something is a shame doesn't mean it's not true, and so far you've produced evidence for your view and I can't for mine, so I can't argue any further.
As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the space much easier.
Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.
Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.
> Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.
Oh, by no means do I object to calling an updated and generalised version by the name of the person who originated the subject! Since you've brought up Galois, I hardly think that he'd recognize the modern theory of Galois connections, but I think that the name is wholly appropriate.
No, what I thought was a shame is if the original theory doesn't get discussed at all. If my only exposure to Peano's work was, for example, the axiom schema in Enderton, then I don't think I'd be able to appreciate why it's such a big deal. That would feel to me like teaching the theory of Galois connections without ever saying anything about field theory! Whereas, on the other hand, I did immediately understand as an undergraduate the magic of being able to define everything in terms of the pointed set using induction, and I think I'd appreciate even more having seen that first and then seeing how it is updated for modern mathematical logic.
In fact, at a casual glance, I still don't see why L1, L3, and the A, M, and E axioms can't be omitted in the presence of the axiom(s) on p. 269, which has been the whole substance of my objection. I believe that there's an answer, but, if I don't see it as a professional mathematician (though not a logician), then surely it can't be true that every undergraduate will appreciate it!
I think from a logic standpoint this also makes sense -- getting to undecidability quickly makes taking the direct route through first-order logic more appealing.
If I'm being honest, I now do feel a little bit deprived, I probably would have enjoyed the categorical view when I was learning this too.
Ah, good point that this was the actual source o# the discussion. This one at least can be argued, because the question is about how things should be axiomatized/defined, not how they are. And certainly the theory of the "natural numbers starting with 1" can be axiomatised just as well as the "natural numbers starting with 0." All these axioms are made by humans, and an appeal to existing axioms here can only say what's been done, not what should be. (And I say this as someone who does start my naturals at 0.)
The original formulation of Peano started at 1.
I never realized it was controversial. I think I've always included 0 in the nat numbers since learning to count.
But there are some programming books I've read, I want to say the Little Typer, or similar, that say "natural number" or "zero". Which makes actually confuses me.
Just like a negative numbers, it's a higher-level abstraction or a model, not a direct observation from the Nature
Likewise, the digit "0" originating from the Hindu-Arabic numeral system[1] is merely a notation, not a number
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1. https://en.wikipedia.org/wiki/Hindu%E2%80%93Arabic_numeral_s...
From one point of view, zero never appearing in nature is exactly an example of it appearing in nature!
From another point of view, do you not think a prairie dog has ever asked another prairie dog, "how many foxes are out there now?" with the other looking and replying "None! All clear!"? Crows can count to at least 5, and will count down until there are zero humans in a silo before returning to it. Zero influences animal behavior!
From a third point of view, humans are natural, so everything we do appears in nature.
From a fourth point of view, all models are wrong, but some models are useful. Is it more useful to put zero in the natural numbers or not? That is: if we exclude zero from the natural numbers, do we just force 90% of occurrences of the term to be "non-negative integers" instead?
type PrairieDogFoxCount = NoFoxesAllClear | SomeFoxes 1..5 | TooManyFoxes
type CrowCount = Some 1..5 | UpsideDown 5..1
type HumanProgrammerCount = 0..MAXINT
type HumanMathematicianCount = 0..∞
> From a third point of view, humans are natural, so everything we do appears in nature.
using this definition everything is Natural, including fore example Complex numbers, which is obviously incorrect, and thus invalidates yr argument
> From a fourth point of view, all models are wrong, but some models are useful. Is it more useful to put zero in the natural numbers or not? That is: if we exclude zero from the natural numbers, do we just force 90% of occurrences of the term to be "non-negative integers" instead?
all models are wrong, but some are really wrong
If all u care is the length of the terms, i.e. "Natural" vs "non-negative integers", then what's wrong with 1-letter set names, like N, W, Z ?
I think the usefulness of including 0 into the set of natural numbers is that it closes the holes in various math theories like [1,2]
1. https://en.wikipedia.org/wiki/Peano_axioms
2. https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
No, that's not "obviously incorrect", nor does it invalidate my argument: that is my exact argument. Complex numbers appear in electromagnetism, in exactly the same sense of "appear", as whole numbers appear in herds of sheep. Which is to say, it's the simplest and most useful model of the situation. And what's more natural than one of the four fundamental forces of nature? And the weak & strong nuclear forces have even more esoteric math structures appearing in their most parsimonious models as well.
> "No Foxes - All Clear" is not the same thing (the same level of abstraction) as 0.
In your model. In my model, it is the same thing. All models are wrong; some models are useful. Which one is more useful? Almost always, the one with 0 as a natural number. What about this:
type PrairieDogFoxCount = NoFoxesAllClear | JustOneFox | ACoupleOfFoxes | SeveralFoxes 3..5 | ManyFoxes
Except you’re wrong here; should we thus call your argument “obviously incorrect”?
Complex numbers are natural; they’re fundamental in quantum mechanics. Ever since Schrödinger’s equation fundamentally required them for time evolution of states, physicists (and philosophers) wondered if they could be removed. Recent experiments say “no.” QED and QFTs are the most precise theories known in all of science.
https://physicsworld.com/a/complex-numbers-are-essential-in-...
Your repeated, willful ignorance on a topic, especially when shown to you, is why you have such low understanding of the incorrect claims you make.
Take a moment and learn. Then maybe you’ll not repeat claims shown to be wrong.
Now go do your homework. Attaching idiot phrases like complex apples is as stupid as claiming we don’t see radio waves so they can’t exist or that matter cannot be mostly empty space because you can stack books.
Your limited imagination, understanding, and unwillingness to learn, even when given a source and phrases to look into, doesn’t apply to those scientists that have done the work.
Any references?
I observe zero.
I don't think zero is an absence of quantity. I don't think zero is the null set.
You can write types in a programming language, but there are other type theory books that do include zero in the natural numbers. And type theory comes from number/set theory. So it's ok if you decide to exclude it, but this is just as arbitrary.
In fact I'd be happy to write `>=0` or `>0` or `=0` any day instead of mangling the idea of zero representing 0 and zero representing something like `None`, `null` or any other tag of that sort. I don't think the natural world has anything like "nothing" it just has logical fallacies.
zero is the cardinality of the empty set
> I observe zero.
it cannot be observed directly at any static point in time, but it can be observed as a dynamic process when some quantity goes down to empty and back up over time
> In fact I'd be happy to write `>=0` or `>0` or `=0` any day instead of mangling the idea of zero representing 0 and zero representing something like `None`, `null` or any other tag of that sort. I don't think the natural world has anything like "nothing" it just has logical fallacies.
N, W, R, etc. - r just well-known names for sets of numbers, nothing stops us from defining better or additional names for them (with self-describing names)
We can discuss Empty type[1] vs Unit type[2], but I think it goes off-topic
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