Zorn's theorem (in various contexts)

Zorn's theorem states that in a partially ordered set—where some elements can be compared—if every chain (a totally ordered subset) has an upper limit, then there is at least one maximal element that cannot be extended further. In simple terms, if you keep extending chains without limit, Zorn's theorem guarantees the existence of a biggest possible element within the set. This principle is fundamental in many areas of mathematics, such as proving the existence of bases in vector spaces or maximal ideals in algebra, where explicit construction is difficult but existence can be assured through this logic.