One of the most useful tools in mathematics.

In a partially ordered set no need be that two elements are always comparable. Meaning not related to each other. A

**chain** is a subset of the partially ordered set such as any two elements are comparable to eachother. Another was of looking at the chain as a subset of the ordering relation because everything in an ordering relation is comparable and only those elements can be related to its other elements.

A upper bound of a chain

is an element such as

for all elements in the chain.

Zorn's Lemma states that in a partially ordered set such as every chain has an upper bound there exists a maximal element

. Meaning If

then

.

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Why is it useful?

For example, without it we would never have Analysis. Because in analysis the fundamental property of real numbers is that if there exists a increasing sequence having an upper bound then it must have a minimal upper bound. This is almost the very definition of Zorn's Lemma.