Linear extension: Difference between revisions

Revision as of 01:10, 5 March 2018

Revision as of 17:27, 16 May 2018

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| title = The Independence of the Prime Ideal Theorem from the Order-Extension Principle

| volume = 64

| jstor = 2586759}}| citeseerx = 10.</ref>1.1.54.8336

}}.</ref>

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 [[model theory|models of set theory]] in which the ordering principle holds while the order-extension principle does not.<ref>{{citation