List of Order Types

This is a list of order types, sorted by the set inclusion relation and by cardinality.

Finite
These ordinals have a cardinality of less than aleph null.
 * $$0$$ Order type zero
 * $$1$$ Order type one
 * $$2$$ Order type two
 * $$3$$ Order type three
 * $$5$$ Order type five
 * $$7$$ Order type seven
 * $$13$$ Order type thirteen

Countably Infinite
These ordinals have a cardinality equal to aleph null.
 * $$\omega$$ Order type omega
 * $$\omega + 1$$ Order type omega plus one
 * $$\omega + 2$$ Order type omega plus two
 * $$\omega 2$$ Order type omega times two
 * $$\omega 2 + 1 $$ Order type omega times two plus one
 * $$\omega 3$$ Order type omega times three
 * $$\omega^2$$ Order type omega squared
 * $$\omega^2 + 1$$ Order type omega squared plus one
 * $$\omega^2 2$$ Order type omega squared times two
 * $$\omega^3$$ Order type omega cubed
 * $$\omega^3 + \omega$$ Order type omega cubed plus omega
 * $$\omega^4$$ Order type omega to the fourth
 * $$\omega^\omega$$ Order type omega to the omega
 * $$\varepsilon_0$$ Order type epsilon naught

Non-Ordinal, Aleph Null

 * $$\omega^*$$ Order type omega star
 * $$\omega^* + \omega$$ Order type omega star plus omega
 * $$\eta$$ Order type eta
 * $$1 + \eta$$ Order type one plus eta
 * $$\eta + 1$$ Order type eta plus one
 * $$1 + \eta + 1$$ Order type one plus eta plus one
 * $$\alpha_p$$ alpha order types where p has a cardinality greater than zero and less than $$\aleph_1$$.

Uncountable
These ordinals have a cardinality greater than or equal to aleph one.
 * $$\omega_1$$ Order type omega one
 * $$\omega_2$$ Order type omega two
 * $$\omega_3$$ Order type omega three
 * $$\omega_\omega$$ Order type omega omega
 * $$i$$ Order type first inaccessible

Non-Ordinal, Aleph One

 * $$\lambda$$ Order type lambda
 * $$1 + \lambda$$ Order type one plus lambda
 * $$\lambda + 1$$ Order type lambda plus one
 * $$1 + \lambda + 1$$ Order type one plus lambda plus one
 * $$\alpha_p$$ Alpha order types where p has a cardinality of $$\aleph_1$$.
 * $$\alpha^{+}_p$$ Alpha-plus order types where p has a cardinality greater than one and less than $$\aleph_2$$

Non-Ordinal, Greater Than Aleph One

 * $$\alpha_p$$ and $$\alpha^{+}_p$$ where p has a cardinality greater than $$\aleph_2$$