The last post is the WINNER!

[sun rise]

Assumptions are losing.

 

[beenherebeforeagain]

is the cardinality of the set of all countable ordinal numbers, called ω 1 {\displaystyle \omega _{1}}

or sometimes Ω {\displaystyle \Omega }

. This ω 1 {\displaystyle \omega _{1}}

is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 {\displaystyle \aleph _{1}}

is distinct from ℵ 0 {\displaystyle \aleph _{0}}

. The definition of ℵ 1 {\displaystyle \aleph _{1}}

implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ 0 {\displaystyle \aleph _{0}}

and ℵ 1 {\displaystyle \aleph _{1}}

. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ 1 {\displaystyle \aleph _{1}}

is the second-smallest infinite cardinal number. Using the axiom of choice we can show one of the most useful properties of the set ω 1 {\displaystyle \omega _{1}}

: any countable subset of ω 1 {\displaystyle \omega _{1}}

has an upper bound in ω 1 {\displaystyle \omega _{1}}

. (This follows from the fact that the union of a countable number of countable sets is itself countable, one of the most common applications of the axiom of choice.) This fact is analogous to the situation in ℵ 0 {\displaystyle \aleph _{0}}

: every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

That's a losing attempt at using infinities...at least for me, it was a bunch of Xes and what resembles coding symbols...

 

[sun rise]

Yes, copy/paste can sometimes be losing. But I bounce back.

 

[Stevicus]

Bouncing in to another win

 

[sun rise]

And another.

 

[Stevicus]

Your win is out of bounce, so therefore it's a loss. I win.

 

[Revoltingest]

Winning after several days of many meals with noodles.

 

[sun rise]

I used my noodle and won.

 

[Stevicus]

Winning without noodles or poodles.

 

[beenherebeforeagain]

I've had oodles of win

 

[sun rise]

Oodles and oodles of win.

 

[Wu Wei]

noodles for the win

 

[Revoltingest]

Ooodles of noodles for the win.

 

[Stevicus]

Oodles of doodles in the margin for the win.

 

[beenherebeforeagain]

lo mien noodles for the win

 

[Wu Wei]

Noodles.....noodles and more noodles.....win

 

[sun rise]

00dles

 

[Wu Wei]

Poodles

 

[sun rise]

We did that already, loser.

 

[Stevicus]

Oodles of strudels were not needed for this win