Realm of objects constructed from the empty set.
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Realm of objects constructed from the empty set.
We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?
But what is the realm called which contains all sets constructed this way?
Re: Realm of objects constructed from the empty set.
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"?
Re: Realm of objects constructed from the empty set.
PS it would help if you insert commas between your subsets.
Re: Realm of objects constructed from the empty set.
Thus, {{}{{}}} becomes {{},{{}}}, which equates to {0,1}, a set with two members; incidentally constituting a definition of the number 2.
Re: Realm of objects constructed from the empty set.
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 { {{{{}}}}, {{}} }
Re: Realm of objects constructed from the empty set.
You can conceptualise this by looking at it from multiple possible perspectives. Logic. Type Theory. Category Theory.ETHstudent123 wrote: ↑Tue Feb 22, 2022 8:53 pm We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?
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.
Re: Realm of objects constructed from the empty set.
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.
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.
Re: Realm of objects constructed from the empty set.
What explanation (more than my demonstration) are you looking for?
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?
Re: Realm of objects constructed from the empty set.
You're looking for two well-known set-theoretic objects, called respectively V and L.ETHstudent123 wrote: ↑Tue Feb 22, 2022 8:53 pm We all know objects such as the empty set {} or {{}{{}}}
But what is the realm called which contains all sets constructed this way?
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