Why is an empty set considered a set? How is it justifiable?

>>8161209
A set is like a box. An empty set is just an empty box.

It's the collection of no distinct object(s).
Just imagine two baskets. One is empty, the other one has the element {Calculator}.
In one basket is nothing, while the other one has a Calcalator. The empty set is a collection of no objects.

More {} = A \ A.

What's left if you take everything out what you have in it.

>>8161213
>>8161215
But the box is an object whose essence can't be identical to set's. The box won't stop being a box if it's empty because it doesn't mean for a box to be box only if it contains something. The definition of set is supposedly for it to be a collection of objects. Shouldn't it be meaningful to call something a set only when there's something it's a collection of?

>>8161218
No. It's enough to say it's the collection of no objects. That's what the definition is for as well.
It's not against the definition after all.

>>8161209
Many algebraic structures need something thath has idempotence.

>>8161226

https://proofwiki.org/wiki/Empty_Set_is_Subset_of_All_Sets

>>8161221
But isn't that just avoiding the problem? How can a 'collection' exist by itself when it\s gathering of objects? Please redefine what that means as I must clearly be misunderstanding. I'm really not trying to be stubborn I just don't understand how this is not problematic

>>8161218
>The definition of set is supposedly for it to be a collection of objects.
Uhhhhh

>>8161228
Yes I know the proof. It comes from the falsity of antecedent in the implication of the meaning of being a subset. But I'm seeing a problem in what comes before that proof can be given, why can you talk about sets as collections when a collection requires something to be a collection of?

>>8161234
>In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
What's wrong with it

>>8161209
>If a set is a collection of distinct objects
This isn't really true, it's more of a layman explanation you give to retards. Unfortunately though, this incorrect explanation of a set is also why so many retards who can't into logic can't believe that the axiom of choice could be false.

Mathematical practice and research has four that idea to be convenient.

For the naivest notion of "set", only Kripke–Platek set theory should be taken as true. All the non-constructivist ZFC spiel can be adopted or not and people would still call it "set". (See e.g. non-wellfounded set theory)

>>8161209
In ZFC sets are made of sets, this sets are also made of sets, wich are also made of sets etc...
So mathematician asked themselves:
Do we want to be philosopher and achieve nothing or mathematicians and achieve great things?

So we decided that the empty set exist, and started constructing interesting set from there that would give us insight about the nature of the reality and beyond.

>>8161302
Obviously choice can be false, just choose a model of, say, ZF, where it fails.

Choice is true in every model of ZF that represents actual sets, in an ontological sense.

>>8161209
Take your pedophile cartoons back to >>>/a/.

>>8162112
Fuck off.

>>8161209
It is convenient. If not for the empty set then sets would not be closed under set operations.

Imagine that you have the set {a,b}
then you do {a,b} - {a}, you get {b}

then you do {b} - {b} and what do you have? According to your system we would no longer have a set. What if we were doing a more complex operation.

Imagine that you have 3 sets A,B, and C and you fill them arbitrarily with objects you got through another procedure. Then your final operation is

(A-B)U(C)

What if A=B? How can you join a set with not a set?

With the empty set we would obviously have {}UC = C, but that is because now we are working with two sets.

Basically, the empty set allows us to do 'set algebra' without worrying about getting out of the set of all sets.

>>8162968
No you fuck off

>>8161209
Nearly every problem someone has with math boils down to taking some english word in a non-rigorous explanation of the topic and being a pedant.

>>8163106
Just wait until he gets to clopen, I want to see his expression.

>>8161209
>an empty set
>an

Get the fuck out

>>8163133
http://www.grammar.com/a-vs-an-when-to-use/

here u go children

>>8163136
Nope. It THE empty set.

>>8163143
There are many empty sets

>>8163143
Prove its uniqueness. Right here, right fuckin now kiddo. Formal as fuck. End it with a neat little box.

>>8163281
Let A and B denote empty sets. By definition, A has no elements and moreover, B has no elements. Therefore every element of A is an element of B, and every element of B is an element of A. Then by the definition of equality of sets, A=B, completing the proof. [math]\qedsymbol[/math]

>>8163291
Would you say that A is an empty set, and B is an empty set, both equal to THE empty set?

>>8163293
No I would say A and B denote the same set, the empty set.

For example, "the set of all even prime numbers greater than 7" and "the set of all rational square roots of two" are the same set despite different language used to describe them.

>>8163297
Prior to completion of the proof, would you say that A and B each represent "an empty set" - an alternative wording to "Let A and B denote empty sets"?

>>8163298
If I had never seen the uniqueness proof before? Maybe, who cares?. Since I accept the proof, I refer to any set with no elements as the empty set.

Also I think your wording is more confusing. If I wrote that proof pedantically I would start "Let A and B denote sets such that |A| = |B| = 0"

My math professor did once basic mathematical research and went into the industry and is highly valued in the academical world in his field.

He taught as well there's just one empty the thus it's _the_ empty set. Believe me or not. I trust him way more than anonymous shitposters.

>>8161209
>Why is an empty set considered a set?
The set is literally nothing, it only exists because of some shitty axiom that everybody decides to blindly follow like the authoritarians they are.

>>8163331
>The set is literally nothing,

No it's literally not. It's a set with literally nothing for its elements.

>>8163331
>it only exists because of some shitty axiom

This much is true.

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This thread is relevant.
>>8162529

>>8163291
I'm just going to take this opportunity to be obnoxiously pedantic.

What you are referring to is the axiom of extensionality which gives you a way of defining extensional equality of sets. When we say that two sets are extensionally equal we mean that from an external perspective they seem to interact the same way with other objects (even though "internally" they may have different structure). This is in contrast to intensional equality (intensionality). Typically one doesn't talk about intensionality in the context of set theory because it is often counterintuitive to "established" analogies and metaphors about how set theory works.

For a clearer example one can look at functions in the context of computability. In computability all functions are defined out of primitive functions and thus inherently have some structure. You can think of a function in computability as a program that you write. In this sense it's easy to see how different programmers could write different programs (functions) that essentially do the same thing while having different structure (and thus different run times and resource footprints). Here we say that two functions f, and g, are extensionally equal if for all inputs x:
f(x) = g(x)
On the other hand, in order to say they are intensionally equal we have to actually compare their internal structure.

I think if you give Set Theory some clever semantics then you could end up with many different empty sets that are still extensionally equal.

>>8161937
>that represents actual sets, in an ontological sense.
I have no idea what you mean.