Linear Extension - Order-extension Principle

The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski in 1930. Marczewski writes that the theorem had previously been proven by Stefan Banach, Kazimierz Kuratowski, and Alfred Tarski, again using the axiom of choice, but that the proofs had not been published.

In modern axiomatic set theory the order-extension theory is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem, but the reverse implication is not provable.

Applying the order-extension principle to a partial order in which every two elements are incomparable shows that (under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the ordering principle, OP, and is a weakening of the well-ordering theorem. However, there are models of set theory in which the ordering principle holds while the order-extension principle does not.