Realm of objects constructed from the empty set.

[ETHstudent123]

We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?

[alan1000]

Given the example you supply, I'm tempted to answer, "the natural number line". But "Realm" is not a term I am familiar with. Can you give a definition? By "Realm" do you mean a synonym for "set" or "subset"?

[alan1000]

PS it would help if you insert commas between your subsets.

[alan1000]

Thus, {{}{{}}} becomes {{},{{}}}, which equates to {0,1}, a set with two members; incidentally constituting a definition of the number 2.

[Skepdick]

Thus, {{}{{}}} becomes {{},{{}}}, which equates to {0,1}, a set with two members; incidentally constituting a definition of the number 2.

Says who?

Why do you count like this....

0 {}
1{ {} }
2{ {}, {}}
3{ {}, {}, {} }

Why don't you count like this?

0 {}
1 {{}}
2 {{{}}}
3 {{{{}}}}

And why don't you count like this?

11 { {{}}, {{}} }
31 { {{{{}}}}, {{}} }

[Skepdick]

We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?

You can conceptualise this by looking at it from multiple possible perspectives. Logic. Type Theory. Category Theory.

In Category Theory it's called the initial object
In Type Theory it's called the bottom type; or empty type
In Logic it's called Falsehood (⊥)

You can think of it as a reification of the Logical principle of explosion. Ex falso quodlibet - from falsehood anything follows.
Or in the language of Category Theory: From the initial object anything follows.
Or in the language of Type Theory: From the empty/bottom type anything follows.

Since there's a 1:1 relationship between type theory, logic and category theory some people call it Computational Trinitarianism, but the term is not popular.

https://ncatlab.org/nlab/show/computational%20trilogy

But really - you can just call it whatever you want. The realm of abstract thought. The Construct. Mathematics. Logic.

Up to you.

[alan1000]

Skepdick, I'm trying to rationalise your response, but please forgive me if I get it wrong.

Why do you count like this....

0 {}
1{ {} }
2{ {}, {}}
3{ {}, {}, {} }

Please explain how you arrived at that; it certainly isn't set theory, which forbids multiplication of the null set.

[Skepdick]

Please explain how you arrived at that

What explanation (more than my demonstration) are you looking for?

it certainly isn't set theory, which forbids multiplication of the null set.

That's a contradiction.

If the null set exists then it has already been multiplied by something. It's a unit of the Null set.

But also... If multiplication of the null set is forbiden then why did you multiply it?

Thus, {{}{{}}} becomes {{},{{}}}

[wtf]

We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?

You're looking for two well-known set-theoretic objects, called respectively V and L.

The von Neumann universe V is what you get when you start with the empty set and keep constructing all the sets you can by applying the standard axioms of set theory.

Gödel's constructible universe L is what you get if you do the same thing, but at each stage, instead of taking all the sets you can make from the axioms, you restrict yourself to only adding those sets that are "first order definable with parameters" from the ones you've already got.

Now the question is, is V = L? That is, do these two constructions give the same universe of sets? The statement that V = L is called the axiom of constructibility. It turns out to be independent of the usual axioms of set theory. If you assume V = L then both the axiom of choice and the continuum hypothesis are true.

Most set theorists reject V = L as being too restrictive a universe of sets.

https://en.wikipedia.org/wiki/Von_Neumann_universe

https://en.wikipedia.org/wiki/Constructible_universe

https://en.wikipedia.org/wiki/Axiom_of_constructibility