User blog comment:Wythagoras/Bounds for Rayo (not my work)/@comment-5150073-20141017075325/@comment-1605058-20141017124837

@Iko, I don't think it's all that reasonable to say, because I have no estimate on how large formulas will get. It might be the case that we will need, say, 3000 symbols to define BB in full formalizm, while exponentiation will only need 800, but going for the former since the start will quite certainly give us better results.

@Hypcos, language of FOST, out of logical connectives, gives us only negation, conjunction and existential quantification.

You can define w as follows: empty set is in w, and for every set a in w, a U {a} is in w too. Moreover, every set with that property has w as a subset.

Alternative definition which was given in the past is: empty set is in w, and a is in w iff there is b in w such that a=b U {b}. I'm not entirely sure if this definition gives us only one set, though.