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Is zero a natural number?
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Is zero a natural number?
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>Is zero a natural number?
Yes.
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it's a placeholder for the additive identity.

>>7809373
retard
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>>7809378
You're wrong. It's a natural number.
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>>7809380
no it's not. a natural number is a positive integer (1, 2, 3, 4, ...).
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>>7809387
No.
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>>7809389
Yes.
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>>7809389
I bet you think circles "just exist."
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zero is such an unnatural number. think about it. no matter how you add it to any other number it does nothing. how is that natural? it's clearly and artificial number, because there is nothing in nature that can be counted, such that there's zero.
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>>7809367
Some people start the set of natural numbers, but most don't

If you start the set of natural numbers with 0, you should call it $\mathbb{N}_0$ instead of $\mathbb{N}$ to be sure.
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>>7809409
>Some people start the set of natural numbers
>Some people start the set of natural numbers at zero
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>>7809409
>>7809410
This is pretty much it. It depends on what country you live in. In the UK they usually start natural numbers from 0, in the US they start from 1.

And it also depends of what area you're studying. In number theory it starts from one, in set theory and computer science it starts from 0. There is basically and disagreement between people(big surprise there)
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>>7809401
But absence of presence is a part of natural existence. I can ask you "How many lights are in the room?" And you can say that there are none, or zero. If "zero" didn't occur in nature, I could ask you how many fins/armadillos/pencils you have and you would always have one or more.
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>>7809417
how are you counting though? in what sense is that a number if there is no countability? you are stating "there are none for me to start counting" if i ask you to count how many lights are in a room and you say "i have counted and found only zero."
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>>7809367
Depends, I think CS and people from Algebra and Logic tend to include it, people from other subfields, especially numerics, don't so they don't have do bother with stupid special cases.
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>>7809414
Honestly I don't see why we don't just redefine "natural number" to be "any non-zero positive number" and then just call the set starting with zero "semi-natural numbers" or something.
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>>7809417
>absence of presence is a part of natural existence
It isn't. You have to literally have the capacity to think and reason in the abstract before it makes any bit of sense to consider it.
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Think outside the box. Maybe 0 is actually -inf. :^)
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>>7809401
I agree with your position on the 'is 0 natural' topic, however the "muh feelings" justification is social-science-tier nonsense.
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>>7809423
Pseudonatural.
>>7809428
No, it's the basis of all numbers. It's as much infinity as it is negative infinity as it is one or zero.

Without 1 to measure 2 by, how could you say zero wasn't two?
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>>7809420
You may not think of it as counting intuitively, but think of it like this; how do you know when to keep counting? When looking at six stones on the ground, how do you know when to keep counting and when to stop counting? You simply check if there are stones that you haven't counted. Each time you count an additional stone, you make a check to determine "Is there an uncounted stone?" You make this check at the very beginning of counting as well. You still make this check when I ask you to count stones in a stone-less room. You look around and realize that there are no uncounted stones, but simply by making this check you are "counting."
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>>7809437
>displaying how it has no natural countable properties in nature
>nonsense

what?
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>>7809409
>Some people start the set of natural numbers, but most don't
You've got that mixed up; it's the other way around. The vast majority of real mathematics uses the term "natural numbers" to refer to the version that includes zero, for all the obvious reasons. The one-based version is used mostly in freshman university courses in the US and that's pretty much it.
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>>7809450
He hates that you have emotions, anon. The mathematical overmind needs you to be ice.
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>>7809458
>for all the obvious reasons
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>>7809425
Is it abstract to say "it is dark?" Darkness is simply the absence of light, yet most would not consider darkness to be an abstract thought. Worms can tell the difference between light and dark, and I highly doubt that they can think and reason in the abstract.
Although perhaps I misunderstand what you mean, to be quite honest I'm not entirely sure I understand your phrasing and diction.
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>>7809447
That's still based on the concept of counting. It's not new or interesting to say the naturals correspond to the counting numbers. If I steelman your post I can get "Zero is the integer terminator," out of it, but that's it. You can equally argue infinity is the "terminator" for $\mathbb{N}$, so your logic doesn't even surpass mine: >>7809440
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>>7809440
>le hookyfishyface.jpg

>>7809450
>zero is SUCH an unnatural number
Next you'll be telling me that men dominate maths and physics because they're "manly" and women dominate biology and nursing because they're "girly"

>>7809459
>implying I'm male
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>>7809367
are italians white?
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>>7809471
You asked me how one would "count" an absence of presence as a refutation to my assertion that absence of presence is a natural concept. Now I have provided my case as to why it is still counting and am waiting on you to posit another refutation.
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>>7809461

Whenever you want to count something, you usually map it onto (wait for it) the natural numbers *as a monoid*. Say you want to count the length of a list; this has a codomain of the natural numbers including zero, for the quite obvious reason that the lists form a monoid and counting is a monoid transformation (there is an empty list, with length zero). This pattern happens again and again; whenever you want to count something that has a monoid structure (which LOTS of things have), you almost always have a codomain that includes zero. The natural-numbers-with-zero are the natural structure for counting.

And mathematicians generally understand this, which is why the natural numbers practically always include zero outside freshman courses.
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>>7809465
>Is it abstract to say "it is dark?"
Yes. Each of those three words has millenia of etymology behind it. If you can rephrase the question as a series of grunts, you MIGHT have a point. Until then, everything you've ever thought to say is equivalent to the most complex process in the known universe.
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>>7809496
I'm not the anon that made that request.
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>>7809472
Sorry, my mistake.

It's women that need ice.
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>>7809496
>it is still counting
Then it's a reflection of its own constructive axiom; ie., a circular argument.
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>>7809509
>And mathematicians generally understand this
Is this one of those Scotsman arguments? How do you define "mathematician" here? Because it sounds to me like you're casting a false generality.
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>>7809458
>The one-based version is used mostly in freshman university courses in the US and that's pretty much it.
Why would they teach the one-based version ONLY in freshman university classes? What fucking purpose does that serve?
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>>7809417
>>But absence of presence is a part of natural existence. I can ask you "How many lights are in the room?" And you can say that there are none, or zero. If "zero" didn't occur in nature
no mate. empiricism has no negation nor ''non-being''. those words are pure concepts... displaying only your fantasy of taking the world for what it is not.

the same thing applies to the notion of necessity and to the notion of emptiness.
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>>7809509
>(there is an empty list, with length zero).
only if you choose to believe in the relevance of such list.
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You can have a field with no cows.

You can't have a field with half a cow or a quarter of a cow (alive cows only).

You can't have a field with -1 cows.

Thats why 0 is natural number.
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in set theory it is, in number theory it tends not to be
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>>7809587
I have a list of integers. I want to split it into a list of even numbers, and a list of odd numbers. This is something that I cannot express at all if you don't allow empty lists.
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>>7809409
â„• = {0, 1, 2, ... }
â„•* = {1, 2, 3, ... }
is the proper notation.
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>>7809581
It prevents a certain kind of lapse in cognition that occurs when you try to "upgrade" intelligence too rapidly. American politics makes perfect sense if you look at it as a temporal system.
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>>7809603
Set theory can't do anything until it defines some mappable notion of numbers, so it tends to need zero to be a natural number so it can construct numbers as separate identities.
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>>7809590
>You can have a field with no cows.
That's called a field. You can't multiply by 'fish' and expect to get a real number.
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>>7809652
>You can't multiply by 'fish' and expect to get a real number.

wat
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>>7809618
That's not even remotely accurate. Each list will, by construction, be non-empty. When you have an empty set like that, you've constructed nothing.
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>>7809652
I have two fields, adjacent to each other but fenced off. In the middle of the fence there is a gate that is currently closed. Both fields have a nonzero of cows on them.

I open the gate. Some cows walk from one field to the other.

After this intermingling of cows, I'd like to be able to conclude that the total number of cows on both fields is the same as it was before, even if all cows from one field all moved to the other field and one of the fields is now empty. If you use natural-numbers-with-zero, this is very easy to state: the sum of the number of cows on field one plus the number of cows on field two is the same at the end as it was at the beginning. If you use natural-numbers-without-zero, stating this *quite natural* observation is suddenly a pain in the dick with nasty special cases in the case of empty fields.

If you make a special case for zero, lots of very natural observations regarding counting become notably impractical (just imagine I have five fields instead of two, and there are 2^5 different special cases to account for in your general statement). That's why the set that includes zero is called the NATURAL numbers, and the set that excludes it is a pointless curiosity.
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>>7809661
>That's not even remotely accurate. Each list will, by construction, be non-empty. When you have an empty set like that, you've constructed nothing.
Elaborate, please? I have the list [1, 3, 5] and I want to split it into the list of even numbers in that list, and the list of odd numbers in that list. If lists can be empty, this has the perfectly coherent value of ([], [1, 3, 5]). If lists cannot be empty, this function cannot be defined at all. Please explain what nonempty list of even numbers that occur in [1, 3, 5] you would use "by construction".
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>>7809367

Zero is not a number. It represents an idea. Similiart to infinity.
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>>7809664
the point is that ''having no cows'' is nice for statements which do not connect to empirical facts. but you can be more constructive in saying that the number of cows in the empty field is actually not even defined.
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>>7809667
but then the question is why to you even think that '''to split it into the list of even numbers in that list'' is interesting to do and/or makes sense.
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>>7809652
> a field with no cows is just called "a field"

but i'd call a field with 3 cows on it "a field" too. not every "field" is "a field with no cows", so this notion gives additional information/practicality of communication and is hence useful.
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>>7809669
you know, actually numbers are abstractions/ideas as well. there is no "five" in nature, only our mind allows for the abstraction that a bunch of stones, a bunch of apples and the fingers on one hand might share a common trait, namely being five. the number 5 is still an abstract concept though.

nothing in math is real or natural in the sense that it is connected to reality in a direct, tangible way. the only question is whether objects can be constructed/defined in way such that they are well-defined.
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>>7809367

9/10 fully vindicated troll post since we're up to 50+ replies.

The real answer is that sometimes it is included, and sometimes not, depending on the subject or argument. A pure mathematician will declare very early in whatever article or textbook they've written, whether or not to include zero as a natural number in their work, for discussion.

Funnily enough, in an American high school of all places, we had a perfectly satisfactory fix for this ambiguity, which was abandoned as soon as I'd gone to college: The first sets that we were acquainted with were: Natural numbers {1,2,3...}, WHOLE NUMBERS (the naturals together with zero, {0,1,2,3...}, Z, Q and R.

A mathematician's common response is that the distinction is immaterial, since both sets have the same important properties: a least element, not dense, and so on. But this is very obviously disingenuous since as I've already written, there is regular need to make clear whether zero is included or excluded "as-a-natural" for this-or-that treatment. And as we all know, abbreviation of simple concepts saves valuable time.

There is no good reason not to have a separate, permanent, and unambiguous set of "whole numbers", or whatever you like, which are the non-negative integers, inclusive of zero. This just hasn't been got round to yet. Call it W, or if someone complains on behalf of an existing set denoted W, call it something else.

>>7809669

7/10 troll post.

>>7809712

This post is wrong in every way that it is possible for it to be wrong. Math and number persist independently of human apperception of them, and this is manifested most obviously cases such as one body orbiting another "two", though of course "number" pops up everywhere else as well.
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>>7809664
>I have two fields
Then you have the same category error as multiplying by "fish."
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>>7809367
>Is zero a natural number?
"Yes, it's the only natural number."

Now go back to >>>/x/
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You niggers need google in yo lives
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>>7809975
This. Theres not much of a point in arguing about which of two conventions is "right" when both make sense in the right context and are either perfectly clear just by the context they are used in or can be easily clarified by unambigous notation like $\mathbb{N}_0$ or $\mathbb{N}^*$ if necessary.
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>>7810029
"Right"'s never the issue with you, is it?
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>>7809367
The term "natural numbers" is a meme term that should have been scrapped centuries ago.

>>7809409
>>7809621
>>7810029
>$\mathbb{N}^*$
$\mathbb{N}^+$ desu. The + shows that you're referring to something that isn't nonpositive. The * has potential for confusion; someone could just as easily include 0 in their definition of $\mathbb{N}^*$.

>>7809975
Better than all those 0.999... threads that hit 100+ posts almost all the time and bump limit half the time.
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>>7810049
I prefer $\mathbb{N}^+$ desu*
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>>7810049
>Better than all those 0.999... threads
How can that be? It's the same exact principle.
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>>7810065
At least whether 0 is a natural number is still debatable in the mathematical community. 0.999... threads are just pure bait.
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>>7810075
>still debatable
That's the first time I hear about people not including 0 in their definition of natural numbers. This is really just a very local matter of a weird convention used only in american universities. There is no debate at all in the mathematical community. I've never seen any paper including in its "conventions and notations" section a sentence about whether or not 0 is in â„•. It's always obvious that 0 âˆˆ â„•
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>>7810085
>American
Not from there, my uni prof said that it indeed is still up for debate, just that they had to pick a convention and they just chose to include 0.
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>>7809367
You can actually get most of the same properties of the natural numbers with or without zero. Honestly, it depends much more on your axioms than anything else. Doesn't matter too much. Generally I say that N does not include zero and I also say a set N' is N unioned with the set containing only zero.

If I really have a need for one or the other, I can just pick either.
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>>7809378

what a moron.
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>>7809472
>expecting someone to refer to you with the correct pronouns over 4chan

thanks liberals
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>>7810094
>I can just pick either.
Not without spawning billions of pointless debates it seems.
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>>7809387
>2016
>defining the naturals using the integers
Peano would cry if he met you
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>>7810101
Its just like how my intro proofs class was. Ready to kill myself.

>>7809387
>Integers intentionally defined as an expansion on the naturals
>defining naturals based on integers

Wow. It's like you have no concept of a derivation.
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>>7809367
In the set by itself, 0 is just the base element of the natural numbers. We're not talking about any other object besides a set.
{{},{{}},{{{}}},{{{{}}}},...} Is a set of natural numbers. In this case, "0" would be the element {}. All systems of natural numbers are isomorphic, so two sets of natural numbers, one starting at 1 and the other at 0, are indistinguishable except for sake of notation.
-1 is also a natural number AS LONG AS YOU DON'T CONSIDER GROUPS, FIELDS, ETC. With how you phrased the question, it's totally valid.
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>>7809423
Unnecessary. Just say "positive," not "nonzero positive."
A positive number is necessarily nonzero.
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>>7810109
It's like you have no concept of divergence. Are you still ready to kill yourself?
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>>7810113
This guy gets it.

Thank you anon, this is some good math.
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>>7810124
I was saying that you are right. I can't believe that something as low level as the naturals debate actually spawns this much disagreement. Save me anon.
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>>7810137
<3
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>>7810141
It's the shilly way OP phrased the question that's got so much power behind it. Click the [Archive] link and hover your mouse over the "Replies" box.
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>>7810146
I feel baited, but in a way that I knew was going to happen the moment I clicked on the thread. Its almost like I wanted this to happen all along.

Here I sit, reading a thread arguing over zero's "naturalness" all while doing compass-straightedge proofs for a historical mathematics class I'm taking for an upper-division math credit.

I feel like I've fallen for the math meme. I'm ok with this.

>>7810145
any time anon.
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>>7810156
That's called destiny. When someone else tries to make it happen, it's called fate.
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>>7810164
Shit dude, that's deep. I'll remember that in lecture today.
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>>7810169
Please do! It could use some contextual propagation.
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>>7810027
A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways:

N = {0, 1, 2, 3, ...}

N = (1, 2, 3, 4, ...}

I mean, really?
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>>7809367
A number denotes a certain quantity. Zero represents nothing, the absence of quantity. Therefore it cannot be a number.
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>>7810192
>A number denotes a certain quantity.
Says who?
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>>7809417
Zero is Infinity.
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>>7809423
Sane people don't want to redefine nature to follow muricunts stupid ideas. I mean even the sentence is as retarded american as it can get:
>any non-zero positive number
>any
>except-zero

I mean the reasons of people trying to justify this are out of this world.
>hur durr we only can comprehend zero because we are so smart so its not a natural number.

0 is natural and you should just fuck off.
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>>7810207
That one prof who asserted that real numbers do not exist?

What happened to that meme, anyway?

>>7810230
Just define it for the context you need it, if it doesn't seem practical or easy to communicate using natural numbers according to one definition, use the other.

Or maybe, if you want to show that 0 is natural, argue how 0 has every quality that defines a natural number. (Or someone else could also argue how it does not, i.e. additive identity)
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>>7810192
The fucking Romans called; they want their shitty ideas about numbers back.
Dumbass.
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>>7810234
Samefag, I mean N.J. Wildberger.

>>7810238
> X said Y
> X is old therefore Y must be wrong.
While I do not agree with the argument you quoted, yours isn't better.
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If 0 isn't a natural number, how are 10, 20, 30, 100, etc. natural numbers? It literally means you count 0 ones, tens, etc.
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>>7810308

Because numbers themselves do not rely upon a given society's notational convention to express those numbers. Admittedly, skimming wiki is telling me that various societies got their heads around zero a bit earlier than I thought, even if they were a bit uncomfortable with the idea due to philosophical autism (leave it to philosophy to obfuscate, confuse and be unhelpful, yet again!)

https://en.wikipedia.org/wiki/0_%28number%29

Cuneiform Babylonian numeral got round your problem very easily, being nothing more than glorified tally marks.

https://en.wikipedia.org/wiki/Babylonian_numerals

While we are unironically on the subject of ethnomathematics, I should also point out that there was a society in Oceania which came up with a novel representation of the binary number system, blended with a base-ten. This is notable insofar as it preceded the need for binary in computers.

http://www.pnas.org/content/111/4/1322.full
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>>7809367
No.

Source: http://oeis.org/A000027
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What's the coolest number ever? Id say it's 7.
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>>7809367
If I remember Peano axioms correctly, 0 is a natural number.
But that really depends on where you're working. I don't really think the pedantics of classification of an object that may also gretly differ, depending on the field of stydy, are worthy of discussion.
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>>7810427
The Peano axioms don't explicitly declare "0" to be anything. They only refer to a first element of the set.

A system of natural numbers is a set (N,x,s) such that:
-N is a set, x is an element of N, and s is an injection s:N->N,v|->s(v).
-There does not exist an element v of N such that s(v)=x.
Additionally, but not necessary for this construction,
-if M is a subset of N such that M contains x, and M is closed under s, then M=N.
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you fucking retards and your literal arguments over definitions

just use $\mathbb{N}_0=\{0,1,2,3,...\},\mathbb{N}_+=\{1,2,3,...\}$ and be done with it
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>>7810776
>$\displaystyle \mathbb{N}_0 = \{0,1,2,3,...\}, \mathbb{N}_+ = \{1,2,3,...\}$
>+ as subscript instead of superscript
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guys if i make up a set of numbers is zero in that set, where zero is some number i made up? i know i can just decide but can you guys please just tell me i literally have no brain
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ok
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>>7809975
>>This post is wrong in every way that it is possible for it to be wrong. Math and number persist independently of human apperception of them
get a load of this rationalist-idealist thinking that he is an empiricist.
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>>7810113
you do understand that ''to have the set of the natural numbers'' is not trivial at all and many do not believe in this set, right ?
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>>7811545
>believe in a set
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>>7810368
But why is 6 afraid of 7
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>>7810351
>For some authors, the terms "natural numbers" and "counting numbers" include 0
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>>7810795
>guys if i make up a set of numbers is zero in that set, where zero is some number i made up?
I do not understand your question.
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>>7810113
This guy just presented you a set-theoritic definition of a "number". You can build natural numbers, real numbers and basically whole ordinals theory with this definition. So, this guy is right and deserves a "/thread".
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>>7811545
There is literally a ZFC axiom stating that it exists. What more do you want?
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I find a chicken that has laid 3 eggs. I take those three eggs. Now the chicken has 0 eggs.

Natural number.

There are no clouds outside today and its hot and dry. There is 0% chance of rain.

Natural number.

I analyzed the nutritional value of corn. The fresh corn had an alcohol content of 0.00mL.

Natural number.
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>>7809367
It doesn't fucking matter, define it to be a natural number if it's more convenient than saying "non-negative integers". It really doesn't matter.
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I am OP and sorry for inducing unintended troll. I just wanted to make a question as simple as possible.

If Mathematics is based upon strict definition and logic, I thought the definition of Natural number should have one and only one answer to this. Not like "it's A in this field of Math but it's B in that field of Math.

I could be more specific in my question but if I had to write additional description to my question, I felt that it is because the definition of Natural number is not good (or should I say, strict) enough.
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>>7812742
>If Mathematics is based upon strict definition and logic, I thought the definition of Natural number should have one and only one answer to this. Not like "it's A in this field of Math but it's B in that field of Math.
You're thinking about it in the wrong way.

Look at geometry. You make a "new" geometry by changing axioms and seeing how that affects everything down the line.

It's based off of "strict definitions" in that you define the assumptions you're working with and then seeing what the logical implication of that particular set of assumptions is.
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>>7811851
>I analyzed the nutritional value of corn. The fresh corn had an alcohol content of 0.00mL.

>Natural number.

I analyzed the nutritional value of corn. The fresh corn had an alcohol content of 0.25mL.

Therefore 0.25 is a natural number as well, by your logic.
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>>7812742
https://en.wikipedia.org/wiki/Peano_axioms

Peano axiom number 1: 0 is a natural number.

>However!

While doing math, the natural numbers are not as essential as other such as the integers, rationals or real numbers. You can define the natural numbers to be 1,2,3.. or 0,1,2,3..., it doesn't matter. You could say that they start in 12783 if you want. It doesn't actually matter and you would end up with the same "mathematics" as long as the notion of successor is there.

It's when we give a structure to the numbers when it begins mattering. But you can't give any meaningful structure to the natural numbers no matter whether you include zero or not, so the natural numbers are not interesting and pretty useless, thus there's no need to give them a one, true definition.
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>>7812865
I was just adding that for dramatic effect.

Its still just 0
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>>7811538

You seem to want to undermine what I said by suggesting either that I'm wrong or that I don't know what I'm talking about (both false), without actually doing either one.

Because if you actually had a way of just proving me wrong (you can't), you'd just come right out and do it. But you don't, and you don't like being right for whatever reason, so you're left with indirect sniping.
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>>7809367
It really depends on the definition you're used to. Peano defines 0 as a natural number. The majority of my lecturers are quite militant about 0 not being a natural number and the inclusion of zero is a "perversion brought about by computer scientists."

Personally, I define zero as a natural number as it fits in with the fact that indexing in many programming languages begins at zero.

Anyone aggressively arguing one way or another is a moron.
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>>7809367
The Axiom of Infinity clearly states that 0 belongs to N
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>>7810113
except thats not the proper definition.

The correct definition of the Von Nuemann Natural Numbers is

0 = empty
N+1= N union {N}

So 1 is infact {{}} but 2 is actually { {}. {{}} }

The zermelo construction is autistic and I assume you probably messed up and instead meant to post the von nuemann construction
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Here's a picture of nature.

How many numbers do you see?

...

...

...

exactly.
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>>7813933
So 1 isn't a natural number?
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>>7809401
>zero is such an unnatural number. think about it.
yup
just ask yourself how many unnatural people you would like to know?
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>>7816067
that heretic is arguing zero is the only natural number
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>>7813933

I see: SIX (being geographical features), including ONE (body of water), ONE (manifested as a large mountain), ONE (as a smaller mountain range), ONE (as a range of trees, among which I can easily discern) TWO (trees), or THREE, or... ; (a rectangular array displayed as, according to its header) 900 by 562 dots, posted by a post number with some sick repeating digits in base ten...

"exactly", kid. Furthermore...

>mathematics is rigorous
>they can't even agree on whether zero is such-and-such a number, or how to pronounce $\partial$ , a rather important symbol in their own language

O I Am Laffin