Zero is not a number

>faggot
Why the homophobia?

Because OP is clearly a fag.

The use of faggot here is closer to homophilia than homophobia you cunty n00b.

0 is in the Reals

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math is not a number, but the sum of the order of it's letters is 42.

1/0 doesn't have an answer because it's just asking the wrong question.

well, if you don't want to call it a number, then don't

>1/0 doesn't have an answer because it's just asking the wrong question.
but all other arithmetic on numbers yield a number.

Yeah sure let's get rid of the identity element of addition. Hey, while we're at it, let's get rid of that silly notion called "i". Math is a crock of shit.

Zero is the real omega prime number.

You're silly

>but all other arithmetic on numbers yield a number.
0 isn't the only number with that property. What we're really talking about is whether or not a number has a multiplicative inverse. The multiplicative inverse for a number of a given set is the number which yields the multiplicative identity when multiplied by the first number. The multiplicative identity for a given set is the number for which other numbers can be multiplied with and return themselves as an answer.
For the set of reals, 1 is the multiplicative identity (1*1=1, 2*1=2, etc.). The multiplicative inverse is 1/x for those numbers that have a multiplicative inverse in this set, and as you already know, 0 does not have a multiplicative inverse in this set i.e. there is no such number which can be multiplied by 0 to yield the multiplicative identity of 1.
For the set of clock hours, the multiplicative identity is also 1 (1*1=1, 2*1=2, etc.). The multiplicative inverse works a little differently with this set compared to the reals in contrast. Going back to basics, we remember that multiplication is defined in terms of iterated addition. So with the reals 5*5=25 because 5+5=10, 10+5=15, 15+5=20, and 20+5=25.
But with clock hours, 5*5=1 because 5:00+5 hours=10:00, 10:00+5 hours=3:00, 3:00+5 hours=8:00, and 8:00+5 hours=1:00.
Given these starting premises, you'll find not only do numbers other than 0 not have multiplicative inverses in this set, but in fact there are exactly twice as many numbers in this set that don't have a multiplicative inverse (2,3,4,6,8,9,10, and 12) vs. the quantity of numbers that do have one (1,5,7,11). No matter how many hours you clock-multiply 2,3,4,6,8,9,10, or 12 by, you will never get to 1:00 e.g. 2 clock-multiplied can only ever yield 2,4,6,8,10, or 12 and 3 clock-multiplied can only ever yield 3,6, 9, or 12.

>0 isn't the only number with that property. What we're really talking about is whether or not a number has a multiplicative inverse. The multiplicative inverse for a number of a given set is the number which yields the multiplicative identity when multiplied by the first number. The multiplicative identity for a given set is the number for which other numbers can be multiplied with and return themselves as an answer.
>For the set of reals, 1 is the multiplicative identity (1*1=1, 2*1=2, etc.). The multiplicative inverse is 1/x for those numbers that have a multiplicative inverse in this set, and as you already know, 0 does not have a multiplicative inverse in this set i.e. there is no such number which can be multiplied by 0 to yield the multiplicative identity of 1.
>For the set of clock hours, the multiplicative identity is also 1 (1*1=1, 2*1=2, etc.). The multiplicative inverse works a little differently with this set compared to the reals in contrast. Going back to basics, we remember that multiplication is defined in terms of iterated addition. So with the reals 5*5=25 because 5+5=10, 10+5=15, 15+5=20, and 20+5=25.
>But with clock hours, 5*5=1 because 5:00+5 hours=10:00, 10:00+5 hours=3:00, 3:00+5 hours=8:00, and 8:00+5 hours=1:00.
>Given these starting premises, you'll find not only do numbers other than 0 not have multiplicative inverses in this set, but in fact there are exactly twice as many numbers in this set that don't have a multiplicative inverse (2,3,4,6,8,9,10, and 12) vs. the quantity of numbers that do have one (1,5,7,11). No matter how many hours you clock-multiply 2,3,4,6,8,9,10, or 12 by, you will never get to 1:00 e.g. 2 clock-multiplied can only ever yield 2,4,6,8,10, or 12 and 3 clock-multiplied can only ever yield 3,6, 9, or 12.
tl;dr

The short version is other numbers do the same thing 0 does in other contexts and that doesn't make any of them stop being numbers.

Or to put it another way, by claiming 0 isn't a number you're inadvertently claiming 8:00 isn't an hour.

Addition is closed in [math]mathbb{Z}[/math]. Therefore [math]a, b \in mathbb{Z} \implies a+b \in mathbb{Z}[math]. Particularly [math] a+(-a) \in mathbb{Z} [/math] Then 0 is a number.

GOD DAMN IT.
Addition is closed in [math]\mathbb{Z}[/math]. Therefore [math]a, b \in \mathbb{Z} \implies a+b \in \mathbb{Z}[/math]. Particularly [math] a+(-a) \in \mathbb{Z} [/math] Then 0 is a number.

0εa comes before the closure of addition in a . You're using circular logic.

By the axiom of existence, [math](\exists X)(X = X)[/math]. It's left as an exercise to the reader that this [math]X[/math] is empty, i.e. [math](\forall Y)(Y \not \in X)[/math].
We now make the definition [math]\emptyset :=: 0 := X[/math]. The rest of the natural numbers are left as an exercise for the reader, along with the existence of the first infinite cardinal (using the axiom of infinity).
QED

Define number.

because you post this in every thread

Accessible cardinal.

existence doesn't imply you're the empty set, genius
from existence you use comprehension to get emptyset
[math] 0 := \{x \in X / x \neq x \} [\math]

having a number of something is a colloquialism, not a factual assessment of the number of goats someone possesses.

If someone doesn't have any goats, the numbe of goats they have is zero.

Numbers do not have to have a unique prime factorization.

"0 means nothing" is an opinion, not a fact.

probabilities and odds are synonyms.

Zero is the additive identity of integers.

QED

so, you proved it? good job, undergrad
let me know when you've attempted the second exercise
also your compliment, while flattering, was unprofessional. please abstain from such comments towards your superiors.

{1} exists and is not empty, jackass

are you having some confusion? i thought the exercise was fairly simple. perhaps you did not realize that the existence axiom did not preclude the existence of more than one set at the same time. please try to pay closer attention next time.

/thread
this is how you construct the reals from empty sets.

...what the fuck did you just say, nigger?

>lmao I said that existence guarantees being empty but I'll pretend I didn't in order to save face in a hentai correspondence imageboard
brainlet

you heard me.

>nigger
Why the racism?

please refer to my lecture, attached. i will not entertain much more silliness from such an unprepared student.

motherfucker that post doesn't even come close to motivating anything around the construction of the reals in ZF what the fuck is wrong with you

Heh......... don't you know blacks are inferior?
Pshh...... stick to your baby maths and leave the real computation to the geniuses like me

>this X is empty
>he still doesn't notice

Zero is the representation of what happens after x - x. A number is a written representation of value that can be quantified by scalable terms. For all intensive purposes, 0 is a number. A number manipulated must equal another number, regardless of increasing/decreasing operations.

aw fuck
i take it all back god damn it
it's late fuck me

>A number manipulated must equal another number, regardless of increasing/decreasing operations.
What about 0/0? or 0^0?

Euclides my ass....a number is just a simbol that represents quantity, 0 just represents lack of it, and is valid as a simbol, therefor as a number.

Cease your inane prattle, brainlets. There is no axiom of existence in Z, ZF, or ZFC. There is an axiom of empty set in ST and KP, but those are inferior set theories.

I have yet to stumble upon a set theory that used:

∃x(x=x)

as an axiom. This seems either trivial or ill defined. If an equivalence relation is defined then this axiom is pointless, and this isn't enough information for a well defined equivalence relation.

If you wanted to presume the existence of the empty set, then use:

∃x∀y¬(yεx)

as an axiom. However, the existence of the empty set is demonstrable within ZF/ZFC, which I will leave as an exercise for the reader.

fuck off, we have all seen several equivalent (to ZF) sets of axioms and they all deal with existence of the empty set differently.
"there exists something" and then using comprehension is as valid as "there exists empty", and it's more concise
mostly because then you can mix existence and infinity

>sets of axioms
oh sweetie......... that's not a set..

embarrassing

what is it then, a category of axioms? what are the morphisms?

please
no

dont do this guys

please what?

>believing the axioms form a set
lmfao what are its elements please tell me hun

it's in the meta language. they aren't anything within the theory; they're a bunch of statements in first-order logic about set theory

>doing algebra with your axioms
wait... shit you might be onto something
nobody steal my idea

>lmfao what are its elements please tell me hun
the axioms

please no
>it's in the meta language
"set" is a word in the meta language. "set of axioms" clearly means informal "set" there. please stop.

describe those sets
please, take your're time

well what axioms do you want to use?

>using imprecise and ambiguous language in the context of mathematics
no, you please stop

any of them
tell me what set "[math](\forall X)(X = X)[/math]" is and show me it exists within ZF

i'm telling you to please stop because you're attracting idiots and trying to be super formal with them and it's all a huge shitfest

it's clear that set of axioms can't mean ZF set if you're talking about the axioms of ZF. you don't need to be an autist about it and start a huge shitpost party

>any of them
>tell me what set "(∀X)(X=X)" is and show me it exists within ZF
you seem confused, the claim was that there are sets of axioms, not that all axioms are sets.

perhaps what you meant was the set {(∀X)(X=X)}

>implying the shitfest isn't part of the fun
i'm typing up my homework at 1AM on a tuesday night dude calm down and relax and you might have fun every once in a while

lad, the claim, if you care to read the fucking comment chain, was that the axioms form a set
all i asked you to do was prove that asinine statement but you're clearly to much of a faggot to read or follow chains of thought

>was that the axioms form a set
see >{(∀X)(X=X)}

in order to prove that's a set, so you must prove its "element" is a set
i'm still waiting, user
i hope you'll have a proof ready when i finish masturbating

"Because Peano said so" doesn't have the same shitposting ring to it.

Wrong by definition.

Zero is defined to be a number, you might as well argue that.

Jesus fucking Christ what a boring question to argue about. Get a life OP, learn real math

This is just a symbol game.
You could (should!) define O to be 1.

1 - 1 = ?

undefined, like 1/0

If 0 isn’t a number then it cannot be used like a number there for cannot have an end point, 1x0 should be 1 right? It’s not it’s 0 because you have no 1