Full Idea
The paradox of the largest ordinal (the 'Burali-Forti') is that the class of all ordinal numbers is apparently well-ordered, and so it has an ordinal number as order type, which must be the largest ordinal - but all ordinals can be increased by one.
Gist of Idea
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal
Source
Shaughan Lavine (Understanding the Infinite [1994], III.5)
Book Reference
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.61
Related Idea
Idea 15918 Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]