I usually hear people say that he wrote the algebraic one (i.e. the one that I'm saying is invalid). But I also have no idea where he supposedly wrote it so I can't verify it. For me it's more hearsay.

By the way I wouldn't hold it against him if he did write the proof that way. I'm pretty certain all the logic/model theory that comes into play came long after his death. The surreal numbers certainly did.