Re: how

On 6/21/2024 3:13 PM, Jim Burns wrote:

On 6/21/2024 2:59 AM, WM wrote:

∪{F(1),F(2),F(3),...} = ℕ  ==>  ∪{ } = ℕ .

⋃{} requires nuance.
Vacuously, everything is in each set in {}
Which makes ⋃{} the universal class.
Which is weird.
The convention is that ⋃{} = {}
But, either way, ⋃{} ≠ ℕ

Oops..
I got that wrong.
⋂{} is what I was thinking of.
Nothing is in any of the sets in {}
Nothing is in ⋃{}
Everything is in every set in {}.
Everything is in ⋂{} ...
except that there may be contexts in which
a thing everything is in isn't an option.
I thought there was a convention
to deal with no universal set,
but I can't lay hands on it at the moment.
Perhaps it would be best
treat ⋂{} like 1/0
"No such thing".
All of which is irrelevant to ⋃{}
which is always {}