0 is a natural number

the naturals can still be a monoid even when 0 isn't considered natural

I didn't make my point clear, sorry about that
I was just trying to say that the star notation is a common notation, and therefore it is most common to exclude an element from a set than to include it, hence, when referring to groups and fields, it is not stupid to mention it with all its elements, including the zero
in this case, you could say that [math]\mathbb{N}[/math] is not a group, and you would be right, but nevertheless, it might not be a group but it is still a monoid, a structure close to that of a group
in my defense, I could add that the peano construction of [math]\mathbb{N}[/math] starts from zero but that would say nothing more

literally not, retard, if anything its a semigroup

yes you're right, but not with the familiar monoid structure of [math]\mathbb{N}[/math], in fact the operation is so bizarre that it stands as useless in almost every kind of approach to [math]\mathbb{N}[/math]

>literally not, retard, if anything its a semigroup
Wrong.

>I was just trying to say that the star notation is a common notation, and therefore it is most common to exclude an element from a set than to include it, hence, when referring to groups and fields, it is not stupid to mention it with all its elements, including the zero
But the * doesn't mean 'exclude zero', it means 'include all units'. The * notation being used for general rings, not just fields [i.e. (Z/4Z)* has two elements, not three]
en.wikipedia.org/wiki/Unit_(ring_theory)#Group_of_units

Define natural number, nigga

t. brainlet

Zero is not a number, it's the (logical) opposite. It's what separates numbers.

>"I know this is bait"
>Makes a bait

I will keep that in mind thanks
but I still hold my cause, that zero is natural
well, I think we made ourselves clear didn't we ?

It's a monoid under multiplication

are you claiming that {0,1,......} is a field? this doesn't even form a group under addition.

How do you guys write the nonnegative integers?
I write [math]\mathbb{Z}^+_0[\math] or [math]\mathbb{Z}_{\geq }[\math].

Fuck, I'm too used to LaTeX backslashes.

I write [math]\mathbb{Z}^+_0[/math] or [math]\mathbb{Z}_{\geq }[/math]

[math] \mathbb{N}\cup \{0\}[/math]

[math] \mathbb{N}-1[/math]

>I know this is bait
well memed

>this is to simplify field notation
you are so far beyond retarded it's almost incredible

when we construct the constructible universe V in set theory, we start off with the empty set, the first set that we can prove exists from the Axiom of Existence and the Axiom of Restricted Comprehension

once we have the empty set, which is the "first" set, we can start trying to construct the number systems, starting with [math]\mathbb{N}[/math]
we COULD construct [math]\mathbb{N}[/math] without including the empty set (so that the first natural number would be 1 = {{}}, the set containing the empty set), but that would make 0 not an ordinal, and therefore not a cardinal (once we get around to defining cardinal numbers)
we would then have no way of assigning cardinality 0 to any set, and we would have no way of using 0 for ordinal arithmetic, which is absolutely fucking retarded

therefore, if your head is still firmly on your shoulders and you have more than a room-temperature IQ, you define the natural numbers to include 0.

there's only one LOGICAL and CONSISTENT way to construct N

most number theorists don't consider 0 a natural number
this just reinforces the idea that number theorists are subhuman trash that don't deserve to breed

finite ordinal
or, if you reject the Axiom of Infinity (wildeburger), then a natural number is just an ordinal

[math]\mathbb{N}\setminus\{0\}[/math]

I know this is bait but fuck you piggot

>we COULD construct N without including the empty set (so that the first natural number would be 1 = {{}}, the set containing the empty set), but that would make 0 not an ordinal
Can you elaborate? How does the definition of ordinal depend on the definition of the naturals?

the first infinite ordinal [math]\omega[/math] is the set of all finite ordinals, which we can prove exists using the Axiom of Infinity
so you prove that [math]\mathbb{N}[/math] exists, which you call the set of all naturals (it's the set of all finite ordinals) and then you prove that it is also an ordinal
when you get around to defining cardinals, which are a special kind of ordinals, you can prove that every [math]n \in \mathbb{N}[/math] is a cardinal, which intuitively corresponds to the size of a set with [math]n[/math] elements
if you don't let 0 be a natural number, then it won't be a cardinal, and we will have no way of talking about the cardinality of the empty set

further, if you make 0 not a natural number by explicitly excluding it from [math]\mathbb{N}[/math], which is the set of finite ordinals, then you won't have a way of using 0 for ordinal arithmetic (which is an entire area of research in set theory; look up what Large Cardinals are)

let me explain this in herp derp language.. zero is just another expression of the notion of nothing or nil - natural numbers express the notion of existence or that there "is" a tangible amount - the argument exists only as a question of "is the non-existance of something an enumeration of amount"

>when you get around to defining cardinals, which are a special kind of ordinals, you can prove that every n∈N is a cardinal, which intuitively corresponds to the size of a set with n elements
>if you don't let 0 be a natural number, then it won't be a cardinal, and we will have no way of talking about the cardinality of the empty set
I'm still not following, the aleph numbers are cardinal numbers and are not contained in N, so why can't 0 be a cardinal when N is defined to not contain 0?

the answer being, sure why not, or no if it suits u, mathematics can sometimes be inextricably linked to deep understandings of reality

this is one of those cases - so zero as a contention of debate is so completely moot, that it's almost laughable

the argument is rooted in primitive human drives to categorize understandings in what feel like logical groupings of knowledge, and as such attempt to extricate all metaphysical unknowns as if they don't exist

in other words, it is better to live in contention than to admit the existence of deep unknown

because all of the finite cardinals are finite ordinals
if you don't let 0 be a finite ordinal then it won't be a cardinal
what is so difficult about this to understand?

i think you misunderstand the definition of N
it is the set of all finite cardinals (as i have said numerous times)
if 0 is not in N then you're ultimately changing your definition of what it means to be an ordinal implicitly, and therefore changing what it means to be a cardinal

>piggot
nice, this is good, bring this back

0 is the idea of a fixed point. an idea is a natural phenomena because its a causality.

People used natural numbers for thousands of years before the concept of 0 existed. Of course 0 isn't natural.

>people did physics for thousands of years before quantum mechanics, of course quantum mechanics isn't physics

>>people did physics for thousands of years before quantum mechanics, of course quantum mechanics isn't physics
It's not, quantum mechanics is just wishful thinking

>"people"
Mathematics is simply an abstract representation of correlaries.

Just because people didn't grasp something doesn't it mean it isn't a natural correlation.

Personal incredulity per argumentum ad populum has NOTHING to do with mathematics

I only come here to read these posts.

Yeah, standard multiplication is bizarre and useless

I usually just write something like n=1,2,...