@David Tanzer gave some nice answers to Puzzle 4, where I was asking for examples of preorders and posets:

> Example of poset: Any collection of sets, ordered by the inclusion relation.

Yes indeed! Like this:

We could even go all out and consider the collection of _all_ sets, ordered by the inclusion relation! This collection is too big to be a set. But we can get around that in various ways, e.g. by considering it as a proper class, or using a [universe](https://en.wikipedia.org/wiki/Grothendieck_universe) of "small" sets, which itself is a "large" set.

A closely related example is Ord, the class of all [ordinals](https://en.wikipedia.org/wiki/Ordinal_number). Ordinals form a _totally_ ordered class. Ord starts out like this:

$$ 0, 1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \dots, \omega \cdot 2, \dots, \omega^2, \dots, \omega^3, \dots, \omega^\omega, \dots, \epsilon_0, \dots $$

but it goes on a lot longer. In fact it goes on _longer than anything!_

> Example of poset: Any collection of sets, ordered by the inclusion relation.

Yes indeed! Like this:

We could even go all out and consider the collection of _all_ sets, ordered by the inclusion relation! This collection is too big to be a set. But we can get around that in various ways, e.g. by considering it as a proper class, or using a [universe](https://en.wikipedia.org/wiki/Grothendieck_universe) of "small" sets, which itself is a "large" set.

A closely related example is Ord, the class of all [ordinals](https://en.wikipedia.org/wiki/Ordinal_number). Ordinals form a _totally_ ordered class. Ord starts out like this:

$$ 0, 1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \dots, \omega \cdot 2, \dots, \omega^2, \dots, \omega^3, \dots, \omega^\omega, \dots, \epsilon_0, \dots $$

but it goes on a lot longer. In fact it goes on _longer than anything!_