0 is a natural number

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?