Von Neumann ordinals #
This file works towards the development of von Neumann ordinals, i.e. transitive sets, well-ordered
under ∈
.
Definitions #
ZFSet.IsTransitive
means that every element of a set is a subset.ZFSet.IsOrdinal
means that the set is transitive and well-ordered under∈
. We show multiple equivalences to this definition.
TODO #
- Define the von Neumann hierarchy.
- Build correspondences between these set notions and those of the standard
Ordinal
type.
Transitive sets #
A transitive set is one where every element is a subset.
This is equivalent to being an infinite-open interval in the transitive closure of membership.
Equations
Instances For
Alias of ZFSet.isTransitive_empty
.
Alias of the forward direction of ZFSet.isTransitive_iff_mem_trans
.
The union of a transitive set is transitive.
The union of transitive sets is transitive.
Alias of the forward direction of ZFSet.isTransitive_iff_sUnion_subset
.
Alias of the forward direction of ZFSet.isTransitive_iff_subset_powerset
.
Ordinals #
A set x
is a von Neumann ordinal when it's a transitive set, that's transitive under ∈
. We
prove that this further implies that x
is well-ordered under ∈
in isOrdinal_iff_isWellOrder
.
The transitivity condition a ∈ b → b ∈ c → a ∈ c
can be written without assuming a ∈ x
and
b ∈ x
. The lemma isOrdinal_iff_isTrans
shows this condition is equivalent to the usual one.
- isTransitive : x.IsTransitive
An ordinal is a transitive set.
The membership operation within an ordinal is transitive.
Instances For
The simplified form of transitivity used within IsOrdinal
yields an equivalent definition to
the standard one.
An ordinal is a transitive set of transitive sets.
An ordinal is a transitive set of ordinals.
An ordinal is a transitive set, trichotomous under membership.
An ordinal is a transitive set, well-ordered under membership.
The Burali-Forti paradox: ordinals form a proper class.