Re: Replacement of Cardinality (infinite middle)

On 8/11/2024 2:10 PM, Ross Finlayson wrote:

On 08/11/2024 09:10 AM, Moebius wrote:

[...]

>
How do you see omega
as the second constant after empty set
an inductive set in ZF?
It's definitely not "all" infinity.

ω is defined to be (the set of) all finite ordinals.
In that sense, ω is all of finiteness.
ω is followed by all the transfinite ordinals.
In that sense, ω is not all of infinity.
Nearly none of it, really.
----
U and V are inductive sets.
⋂{ind:U} is the intersection of inductive U.subsets.
⋂{ind:U} is inductive.
for each inductive A ⊆ U:  ⋂{ind:U} ⊆ A ⊆ U
⋂{ind:V} is the intersection of inductive V.subsets.
⋂{ind:V} is inductive.
for each inductive B ⊆ V:  ⋂{ind:V} ⊆ B ⊆ V
In particular, U∩V is an inductive U.subset and V.subset.
As an inductive V.subset,
⋂{ind:V} ⊆ U∩V ⊆ V
As an inductive U.subset,
⋂{ind:U} ⊆ ⋂{ind:V} ⊆ U∩V ⊆ U
⋂{ind:U} ⊆ ⋂{ind:V}
Similarly,
⋂{ind:V} ⊆ ⋂{ind:U}
⋂{ind:U} = ⋂{ind:V}
⋂{ind:U} = ⋂{ind:V} := ⋂{ind}
the unique intersection of inductive subsets.
----
{fin} is the set of finite ordinals.
⋂{ind} is the intersection of inductive subsets.
There is no first finite.ordinal ∉ ⋂{ind}
There is no finite ordinal ∉ ⋂{ind}
{fin} ⊆ ⋂{ind}
Each inductive set ⊇ ⋂{ind}
{fin} is inductive.
{fin} ⊇ ⋂{ind}
{fin} ⊆ ⋂{ind}
{fin} ⊇ ⋂{ind}
{fin} = ⋂{ind} := ω

How do you see omega

ω is the set of all finite ordinals and is,
for each inductive set,
the intersection of inductive subsets.