Dushnik–Miller theorem (original) (raw)

In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable case there exists an order embedding into a proper subset of the given order; however, they provided examples showing that this strengthening does not always hold for uncountable orders.