User blog comment:Wythagoras/All my stuff/@comment-10429372-20130715145943/@comment-1605058-20130715171506

Let say we have some formal theory T with required syntax. Let T-definable be analoguous to Rayo-nameable in obvious way. Let definable mean T-definable for some formal theory T. Because syntax must be finite, there is countably many syntax-es and thus countably many theories, each containing only countably many sentnces. Via pairing argument we have countably many definable objects. From uncountability of ordinals, undefinable ordinals must exist. I think this is good argument.