The order type of the rationals.

Question

Herewith another mind-numbingly naive question from a reader of philosophy.

My question concerns the order type of the rational numbers.

Omega squared seems a natural first choice, but obviously this does not look anything like the natural ordering of the rationals.

Is it known where the order type of Q occurs in the hierarchy of ordinal numbers? Is there a known ordinal-arithmetic expression describing it a function of omega?

Finally, I really must buy a textbook on the subject of Set Theory. Wiki is a fantastic resource and the maths pages are of exceptionally high quality, but I don't want to get into bed at night with my laptop. Is there a standard, undergraduate text that could be recommended.

Answer 1

The ordinals are order types of well-ordered partial orders. The rational numbers are not well-ordered, therefore their order type does not occur within the ordinal hierarchy.

Answer 2

E. Kampke's book on set theory, which I think has a Dover edition, has some material on the order type of the rationals. It's not found among the ordinals because it's not well-ordered. However, there's a proof, which I seem to recall goes back to Cantor, proving that any two countable linearly ordered sets without endpoints that are densely ordered (i.e. between any two points there's another) are order-isomorphic.