Category Theory Illustrated – Types
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Category Theory Illustrated – Types
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Re: Category Theory Illustrated – Types
#2It's a great introduction, but I find the premise a bit funny. It starts with Russell's paradox, insinuates that solving it within set theory makes set theory complex (it doesn't, you basically just restrict what can be used to build a set), and then introduces a system that is fundamentally more complex.
Re: Category Theory Illustrated – Types
#3[flagged]
Re: Category Theory Illustrated – Types
#4[dead]
Re: Category Theory Illustrated – Types
#5Regarding Russell’s paradox, its dual is also interesting: Consider the set D := { s | s ∈ s }, the set of sets that do contain themselves. Does D contain itself? It might or it might not, neither causes a contradiction. Tnis shows that you don’t need an antinomy for a set comprehension to be ill-defined.
Re: Category Theory Illustrated – Types
#6Regarding Russell’s paradox, its dual is also interesting: Consider the set D := { s | s ∈ s }, the set of sets that do contain themselves. Does D contain itself? It might or it might not, neither causes a contradiction. Tnis shows that you don’t need an antinomy for a set comprehension to be ill-defined.
Why is it ill-defined? As you said, there's no contradiction.
Also, in the usual ZF set theory, it's empty.
Re: Category Theory Illustrated – Types
#7Regarding Russell’s paradox, its dual is also interesting: Consider the set D := { s | s ∈ s }, the set of sets that do contain themselves. Does D contain itself? It might or it might not, neither causes a contradiction. Tnis shows that you don’t need an antinomy for a set comprehension to be ill-defined.
Why is it ill-defined? As you said, there's no contradiction. Also, in the usual ZF set theory, it's empty.
... and, as such, it doesn't contain itself!
Re: Category Theory Illustrated – Types
#8> a set can contain itself
Can it?
> a term can have only one type... Due to this law, types cannot contain themselves
Doesn't look like one follows from the other...
Re: Category Theory Illustrated – Types
#9> a set can contain itself Can it? > a term can have only one type... Due to this law, types cannot contain themselves Doesn't look like one follows from the other...
The set of all sets that contain itself ;)