Can Zero Be Considered an Empty Set in Set Theory?

[ferman]

My question would be:
In mathematics of set, can be 23 a set with 23 elements.
Could be 4 a set of four elements?
Then would be zero the empty set.
And if zero is the empty set, would be 3 emtpy sets equal to 1 empty set.

 

[morphism]

Look up ordinals.

 

[HallsofIvy]

In mathematics nothing IS a single thing!

It is certainly possible, and done in some treatices, to DEFINE "0" to be the empty set, then define "1" to be the set whose only member is the empty set, define "2" to be the set whose only members are "0" and "1", etc. That means the "2" is a set containing 2 members and, indeed, "23" would be a set containing "23" members- but not just any such set.

I don't know what you mean by "And if zero is the empty set, would be 3 emtpy sets equal to 1 empty set." 3 sets of anything are not "equal to" (in the strict sense of "are exactly the same thing") a single set of anything- three sets are not the same as one set. Since you are talking about sets, you might mean the union of the sets: in that case, yes, the union of 3 empty sets is indeed the empty set.

Or, since you are talking about numbers, you might mean the sum of the numbers "represented" by three empty sets. In that case you would have to define such a sum. That also can be done and, with the usual definition, yes, again, the sum of "three empty sets", that is 0+ 0+ 0, is, indeed, 0 or the empty set.

 

[ferman]

What I would say is that the mathematics of set is more extensive and complete than the pure mathematics.
This doesn’t mean that mathematic of set and pure mathematics are different, but complementary.
0 is de empty set; a set with no one element. For example, a void room of students.
Two empty sets would be 0 + 0; for example two void room of students.
Now well, in pure mathematics, 3 void rooms of students have 0 students, and 1 void room of students are equal to 3 void rooms of students due to the total number of students is 0.
But in mathematics of set, 3 void o empty room of students a not equal than 1 void o empty room of students due to in mathematics of set we also take in mind the number of set and no alone the number of contained elements.
But much more, in mathematics of set we can operate with empty set and in pure mathematics no.
In the drawing we can see as it is possible operations with empty sets, at least in my opinion.

 

[DeadWolfe]

Another person whos probably working on a perpetual motion machine.

 

[Office_Shredder]

What I would say is that the mathematics of set is more extensive and complete than the pure mathematics.
This doesn’t mean that mathematic of set and pure mathematics are different, but complementary.
0 is de empty set; a set with no one element. For example, a void room of students.
Two empty sets would be 0 + 0; for example two void room of students.
Now well, in pure mathematics, 3 void rooms of students have 0 students, and 1 void room of students are equal to 3 void rooms of students due to the total number of students is 0.
But in mathematics of set, 3 void o empty room of students a not equal than 1 void o empty room of students due to in mathematics of set we also take in mind the number of set and no alone the number of contained elements.
But much more, in mathematics of set we can operate with empty set and in pure mathematics no.
In the drawing we can see as it is possible operations with empty sets, at least in my opinion.

You make the same mistake here... 3 void rooms of students isn't equal to one void room of students, because there are a different number of rooms. You just dressed it up in different terminology, but students and rooms can be exchanged freely with elements and sets and the example is the same

 

[ferman]

This is the questions, terminology but also concepts and physical realities.
Currently we use some disconnection among mathematics of set and pure mathematics.
For example:
--If we divide 36 students among 3 empty rooms, pure mathematics says 36/3=12 and stop here.
--But the reality (in mathematics of set) says: a set A (36 students) is divided by a set B (3 empty rooms) and give us three occupied rooms with 12 students each ones in the following way:
A (36 student) / B (3 empty rooms) = C (room, 12 students), D (room, 12 students) and E (room, 12 students).
This last solution (in set) shows us the total result and not a partial result as in pure mathematics (12).
So, in this example we can see as in set mathematics we can observe more complete results y more exposition of the physical reality.